Research Article Reciprocity and Self-Tuning Relations without...

Research ArticleReciprocity and Self-Tuning Relations without Wrapping

Davide Fioravanti1 Gabriele Infusino23 and Marco Rossi4

1Sezione INFN di Bologna Dipartimento di Fisica e Astronomia Universita di Bologna Via Irnerio 46 40126 Bologna Italy2Dipartimento di Fisica dellrsquoUniversita della Calabria Arcavacata Rende 87036 Cosenza Italy3Laboratoire Jean Alexandre Dieudonne Universite Nice Sophia Antipolis 06100 Nice France4Dipartimento di Fisica dellrsquoUniversita della Calabria and INFN Gruppo Collegato di Cosenza Arcavacata Rende87036 Cosenza Italy

Correspondence should be addressed to Davide Fioravanti fioravantiboinfnit

Received 6 August 2015 Accepted 15 October 2015

Academic Editor Stefano Moretti

Copyright copy 2015 Davide Fioravanti et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

We consider scalar Wilson operators of N = 4 SYM at high spin 119904 and generic twist in the multicolor limit We show thatthe corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certainldquoreciprocityrdquo and functional ldquoself-tuningrdquo relations up to all terms 1119904(ln119904)119899 (inclusive) at any fixed rsquot Hooft coupling 120582 Of coursethis relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same orderin 119904 On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalousdimension expansions at the next order (ln119904)21199042 at fixed rsquot Hooft coupling in such a way to reestablish the aforementioned relation(which fails otherwise)

1 Introduction Aims and Results

One of themajor achievements ofmodern theoretical physicsis the so-called AdSCFT correspondence [1ndash3] and itsdescription in terms of integrability tools [4ndash19] In factbeing a strongweak coupling duality the nonperturbativeexact though not necessarily explicit (as a simple examplewe can mention just with reference to the present paper thatthe following nonlinear integral equation (which governs thespectrum) is not explicitly solvable) nature of integrability isof incomparable value and utility In particular the spectrumof anomalous dimensions of composite operators in N = 4

super Yang-Mills (SYM) theory ought to correspond to theenergy spectrum of states in type IIB superstring theory inAdS

5times S5 and both must be described by an integrable

systemAmong the different sectors of multicolor N = 4 SYM

(perturbatively closed under renormalisation) one of themost studied ones is the so-called sl(2) scalar twist sectorThisis spanned by local composite operators of single trace form

Tr (D119904

Z119871

) + sdot sdot sdot (1)

where D is a (light-cone) covariant derivative acting in allthe possible ways on the 119871 complex bosonic fields Z thetrace ensuring gauge invariance The Lorentz spin of theseoperators is 119904 and 119871 coincides with the twist that is theclassical dimensionminus the spinTheAdSCFT correspon-dence relates operators (1) to spinning folded closed stringson AdS

5times S5 spacetime with AdS

5and S5 angular momenta

119904 and 119871 respectively [20 21]One of the several reasons for the large interest in these

operators is their similarity with twist operators in QCDwhere maybe the scalars are substituted by fermions thatis the quarks or gauge fields because of integrability inN = 4 these cases would be dealt with in an analogousmanner [22 23] Similarities among the two theories give thepossibility to believe thatQCDcould takemany advantages ofa full all-loop solution of its supersymmetric counterpart InQCD in the framework of Partonic Model the Lorentz spin119904 is the conjugated variable in the Mellin transform (of thesplitting function for instance which gives the anomalousdimension) to the Bjorken variable 119909 namely the fractionof the hadron momentum carried by the single parton (ofcourse the coupling does run in QCD unlike what happens

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 762481 21 pageshttpdxdoiorg1011552015762481

2 Advances in High Energy Physics

in the maximally supersymmetric theory) In this contexttwo regimes emerge naturally 119909 rarr 0 governed by theBFKL equations [24] and 119909 rarr 1 corresponding exactlyto large values of the Lorentz spin 119904 rarr infin Properties ofthis second (called quasielastic) regime can be deduced bylarge spin results in three-loop twist 2 QCD calculationsIn particular we can highlight two main features aboutanomalous dimension of twist operators

(1) The leading term has a logarithmic scale

120574 (119904) sim ln 119904 119904 997888rarr infin (2)

(2) Subleading terms obey hidden relations the Moch-Vermaseren-Vogt constraints [25 26] in brief termsproportional to ln 119904119904 and 1119904 are completely deter-mined by terms proportional to ln 119904 and 119904

0 Theseconstraints are related with spacetime reciprocity ofdeep inelastic scattering and its crossed version of119890+

119890minus annihilation into hadrons

N = 4 gauge theory shares at large 119904 these features andbesides allows us an understanding of their origin and thuspossible extension to QCD In specific the asymptotic large119904 series of the anomalous dimensions are believed to beconstrained by nonperturbative (in 120582 = 8120587

2

1198922 the rsquot Hooft

multicolor coupling) functional relations that work for anyfinite value of the twist 119871 To be more precise conformalsymmetry implies that anomalous dimensions 120574(119892 119871 119904) oftwist operators are functions of the conformal spin thistranslates into the following ldquoself-tuningrdquo functional relation[27 28]

120574 (119892 119871 119904) = 119875 (119904 +

1

2

120574 (119892 119871 119904)) (3)

Additionally this has to be meant asymptotically in the sensethat the function (the function 119875 in (3) actually depends onthe twist of the operator as well)

119875 (119904) =

infin

sum

119899=0

119886(119899)

(ln119862 (119904))119862 (119904)

2119899 (4)

is represented by a series in 119904 via the conformal Casimir

119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) (5)

only Relation (4) is equivalent to the so-called reciprocitysymmetry 119909 rarr minus119909 but for the Mellin space variable 119904 andan important piece of information is that the function 119886(119899) hasthe form of an upper truncated Laurent series

119886(119899)

(ln119862 (119904)) =119872

sum

119898=minusinfin

119887(119899)

119898(119892 119871) (ln119862 (119904))119898 119872 lt infin (6)

that is 119886(119899)(ln119862(119904)) depends on 119904 only through powers ofln119862(119904) For twist two and three negative powers of ln119862(119904) areabsent and 119886(119899) is a polynomial for generic twist howeverone has to cope with the infinite Laurent series (6)

Relations (3) and (4) both in QCD and in N = 4 SYMare developed checked in various cases and discussed in [27ndash37] Recently they have been proven restrictively to twist twooperators but in a generic conformal field theory in [38]with some arguments for their validity in nonconformal the-ories at the end Clearly they provide important informationon the high spin expansion of anomalous dimension of twistoperators UnlikeQCD inN = 4 SYM it is possible to obtainbetter and more suitable results as we can consider theserelations into the framework of integrability The latter wasfirstly discovered in the planar limit for the purely bosonicso(6) sector at one loop [4] then it was extended to all thegauge theory sectors and to all loops [5ndash10] In specific itwas found that every composite operator can be thoughtof as a state of a ldquospin chainrdquo whose Hamiltonian is thedilatation operator itself although the latter does not havean explicit expression of the spin chain form but for thefirst few loops Nevertheless the spectrum of infinitely longoperators has turned out to be exactly described by a set ofAsymptotic Bethe Ansatz (ABA) equations [5ndash10] On theother hand anomalous dimensions of operators with finitequantum numbers depend not only on ABA data but alsoon finite size ldquowrappingrdquo corrections [11 12] Subsequentprogress has shown that a set of Thermodynamic BetheAnsatz (TBA) equations [13ndash17] or an equivalent 119884-systemof functional equations [18] together with certain additionalinformation [19] provides a solid ground for exact (anylength any coupling) predictions on anomalous dimensionsof planarN = 4 SYM

Despite this impressive progress we believe that it isstill important to define the largest domain of compositeoperators for which the ldquosimplerrdquo ABA equations give thecorrect anomalous dimensions especially in connexion withotherwell-established relevant equations In fact we intendedthis to be the main aim of this paper and the most naturalsetting to perform this study to be the reformulation of ABAequations as one (Non)linear Integral Equation (NLIE) [39]

Generically and sketchily for operators composed of119871 elementary fields ABA gives the correct perturbativeexpansion of the anomalous dimension up to 119871 minus 1 loopsStarting from 119871 loops ldquowrappingrdquo diagrams which are nottaken into account by ABA start to contribute In thisgeneral framework the high spin limit of fixed twist operatorsseems to offer a better scenario Perturbative (up to sixloops) computations [40ndash42] for short (twist two and three)operators show that wrapping diagrams (which enter fromfour loops on) actually give contributions which in the highspin limit behave as 119874((ln 119904)21199042) It is then natural to ask ifsuch property extends to higher (and possibly to all) ordersof perturbation theory In this paper we want to provideevidence in favour of this picture by using the self-tuningand reciprocity properties In order to do that we first rewrite(Section 2) the ABA equations as NLIEs for the countingfunction Then in Section 3 we specialise ourselves to theminimal anomalous dimension state and go to the high spinlimit while keeping the twist finite upon computing thepositions of the external holes and the effect of the nonlinearterms we write a linear integral equation equivalent to ABA

Advances in High Energy Physics 3

up to the orders 1119904(ln 119904)119899 119899 isin Z 119899 ge minus1 (inclusive) InSection 4 we use this linear integral equation to computeat the same order of 119904 but at all values of the couplingthe ABA prediction for the minimal anomalous dimensionThen in Section 5 we show the latter to satisfy the self-tuningand reciprocity relations Interestingly we also find that thesolution of the linear integral equation respects suitable self-tuning and reciprocity relations (up to this order in 119904) Finallywe provide some arguments supporting the idea that at highspin wrapping corrections affect twist operators starting fromorders (ln 119904)21199042 so that self-tuning and reciprocity relationsstill hold (and likely also a modified (non)linear integralequation)

2 From the ABA to the NLIE

As planned in the Introduction we start from the ABAequations [5ndash10] for the sl(2) sector ofN = 4 SYM

(

119906119896+ 1198942

119906119896minus 1198942

)

119871

(

1 + 1198922

2119909minus

(119906119896)2

1 + 11989222119909

+(119906

119896)2)

119871

=

119904

prod

119895=1

119895 =119896

119906119896minus 119906

119895minus 119894

119906119896minus 119906

119895+ 119894

(

1 minus 1198922

2119909+

(119906119896) 119909

minus

(119906119895)

1 minus 11989222119909

minus(119906

119896) 119909

+(119906

119895)

)

2

sdot 1198902119894120579(119906119896 119906119895)

(7)

where

119909plusmn

(119906119896) = 119909 (119906

119896plusmn

119894

2

)

119909 (119906) =

119906

2

[

[

1 +radic1 minus

21198922

1199062

]

]

120582 = 81205872

1198922

(8)

120582 being the rsquot Hooft coupling The so-called dressing factor[10 43 44] 120579(119906 V) is given by

120579 (119906 V) =infin

sum

119903=2

infin

sum

]=0120573119903119903+1+2] (119892)

sdot [119902119903(119906) 119902

119903+1+2] (V) minus 119902119903 (V) 119902119903+1+2] (119906)]

(9)

the functions 120573119903119903+1+2](119892) = 119892

2119903+2]minus1212minus119903minus]

119888119903119903+1+2](119892) being

120573119903119903+1+2] (119892)

= 2

infin

sum

120583=]

1198922119903+2]+2120583

2119903+120583+] (minus1)

119903+120583+1 (119903 minus 1) (119903 + 2])2120583 + 1

sdot (

2120583 + 1

120583 minus 119903 minus ] + 1)(

2120583 + 1

120583 minus ])120577 (2120583 + 1)

(10)

and 119902119903(119906)

119902119903(119906) =

119894

119903 minus 1

[(

1

119909+(119906)

)

119903minus1

minus (

1

119909minus(119906)

)

119903minus1

] (11)

being the expression of the 119903th charge in terms of the rapidity119906 Operators (1) of twist 119871 correspond to zero momentumstates of the sl(2) spin chain described by an even number119904 of real Bethe roots 119906

119896which satisfy (7) For a state described

by the set of Bethe roots 119906119896 119896 = 1 119904 the eigenvalue of

the 119903th charge is

119876119903(119892 119871 119904) =

119904

sum

119896=1

119902119903(119906

119896) (12)

In particular (asymptotic) anomalous dimension of (1) is

120574 (119892 119871 119904) = 1198922

1198762(119892 119871 119904) (13)

Let us focus (in this section from Section 2 on we willrestrict to the minimal anomalous dimension state) on statesdescribed by positions of roots which are symmetric withrespect to the originThese are in particular zero momentumstates An efficient way to treat states described by solutionsto a (possibly large) number of (algebraic) Bethe Ansatzequations consists in writing one nonlinear integral equationcompletely equivalent to them (cf [45] and references thereinfor the idea without holes degree of freedom) The nonlinearintegral equation is satisfied by the counting function 119885(119906)which in the case (7) reads as

119885 (119906) = Φ (119906) minus

119904

sum

119896=1

120601 (119906 119906119896) (14)

where

Φ (119906) = Φ0(119906) + Φ

119867(119906)

120601 (119906 V) = 1206010(119906 minus V) + 120601

119867(119906 V)

(15)

with

Φ0(119906) = minus2119871 arctan 2119906

Φ119867(119906) = minus119894119871 ln(

1 + 1198922

2119909minus

(119906)2

1 + 11989222119909

+(119906)

2)

1206010(119906 minus V) = 2 arctan (119906 minus V)

120601119867(119906 V)

= minus2119894 [ln(1 minus 119892

2

2119909+

(119906) 119909minus

(V)1 minus 119892

22119909

minus(119906) 119909

+(V)

) + 119894120579 (119906 V)]

(16)

It follows from its definition that the counting function 119885(119906)is a monotonously decreasing function In addition in thelimit 119906 rarr plusmninfin since

120601 (119906 V) + 120601 (119906 minusV) 997888rarr plusmn2120587 minus

4

119906

+

21198941198922

119906

(

1

119909minus(V)

minus

1

119909+(V)

)

+ 119874(

1

1199063)

(17)


one has the asymptotic behaviour

119906 997888rarr plusmninfin

119885 (119906) 997888rarr ∓ (119871 + 119904) 120587 +

119871 + 2119904 + 120574 (119892 119871 119904)

119906

+ 119874(

1

1199063)

(18)

Thismeans that there are 119871+119904 real points 120592119896such that 119890119894119885(120592119896) =

(minus1)119871+1 It is a simple consequence of the definition of 119885(119906)

that 119904 of them coincide with the Bethe roots 119906119896 For Bethe

equations (2) Bethe roots are all real and are all contained inan interval [minus119887 119887] of the real line The remaining 119871 pointsare called ldquoholesrdquo [39 46ndash52] they also are real and theywill be denoted as 119909

ℎ One should distinguish between 119871 minus 2

ldquointernalrdquo or ldquosmallrdquo holes 119909ℎ ℎ = 1 119871 minus 2 which reside

inside the interval [minus119887 119887] and two ldquoexternalrdquo or ldquolargerdquo holes119909119871minus1

= minus119909119871 with 119909

119871gt 119887

We finally remark that anomalous dimension appears (18)in the limit 119906 rarr infin of the counting function We will comeback to this fact in Appendix A

As we are in presence of holes we may follow theextension of the idea as developed in [53] andmake use of theCauchy theorem to obtain a simple integral formula (1198851015840

(V) =(119889119889V)119885(V) cf also [54] for more details on the followingformulae)

119904

sum

119896=1

119874 (119906119896) +

119871

sum

ℎ=1

119874 (119909ℎ)

= minusint

+infin

minusinfin

119889V2120587

119874 (V) 1198851015840

(V)

+ int

+infin

minusinfin

119889V120587

119874 (V)119889

119889VIm ln [1 + (minus1)119871 119890119894119885(Vminus1198940

+)

]

(19)

Application of (19) to the derivative of (14) gives

1198851015840

(119906) = Φ1015840

(119906) + int

+infin

minusinfin

119889V2120587

119889

119889119906

120601 (119906 V) 1198851015840

(V)

+

119871

sum

ℎ=1

119889

119889119906

120601 (119906 119909ℎ)

minus int

+infin

minusinfin

119889V120587

119889

119889119906

120601 (119906 V)119889


+)

]

(20)

We introduce the notations

120590 (119906) = 1198851015840

(119906)

1198711015840

(119906) =

119889

119889119906

Im ln [1 + (minus1)119871 119890119894119885(119906minus1198940+)

]

(21)

and pass to Fourier transforms 119891(119896) = int

+infin

minusinfin

119889119906119890minus119894119896119906

119891(119906)keeping in mind that

Φ0(119896) = minus

2120587119871119890minus|119896|2

119894119896

Φ119867(119896) =

2120587119871

119894119896

119890minus|119896|2

[1 minus 1198690(radic2119892119896)]

1206010(119896) =

2120587119890minus|119896|

119894119896

120601119867(119896 119905) = minus8119894120587

2119890minus(|119905|+|119896|)2

119896 |119905|

[

infin

sum

119903=1

119903 (minus1)119903+1

119869119903(radic2119892119896) 119869

119903(radic2119892119905)

sdot

1 minus sgn (119896119905)2

+ sgn (119905)infin

sum

119903=2

infin

sum

]=0119888119903119903+1+2] (119892) (minus1)

119903+]

sdot (119869119903minus1

(radic2119892119896) 119869119903+2] (radic2119892119905) minus 119869119903minus1 (radic2119892119905) 119869119903+2] (radic2119892119896))]

(22)

We obtain the equation

(119896)

=

119894119896

1 minus 119890minus|119896|

Φ (119896) minus 2

119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(23)

and for 119885(119896) the equation

119885 (119896)

=

1

1 minus 119890minus|119896|

Φ (119896) minus 2

119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

1

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) 119894119905 [

119885 (119905) minus 2 (119905)]

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(24)


which is the nonlinear integral equation for the countingfunction 119885(119906) describing states of the sl(2) sector We willfind it convenient to introduce the following function

119878 (119896) =

sinh (|119896| 2)120587 |119896|

(119896) +

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

120587119871

sinh (|119896| 2)(1 minus 119890

minus|119896|2

)

minus

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

[cos 119896119909ℎminus 1]

(25)

because in Appendix A we show that it satisfies the simplerelation

lim119896rarr0

119878 (119896) =

120574 (119892 119871 119904)

2

(26)

The function (25) satisfies the nonlinear equation

119878 (119896) =

119871

|119896|

(1 minus 1198690(radic2119892119896)) +

119894119896

1 minus 119890minus|119896|

sdot int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

sdot [

sum119871

ℎ=1(cos 119905119909

ℎminus 1) + 119871 (1 minus 119890

minus|119905|2

) minus (119894119905120587) (119905)

1 minus 119890minus|119905|

]

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

|119905|

2 sinh (|119905| 2)119878 (119905)

(27)

Now the introduction of the ldquomagic kernelrdquo [10]

(119905 1199051015840

) =

2

1199051199051015840[

infin

sum

119899=1

119899119869119899(119905) 119869

119899(1199051015840

)

+ 2

infin

sum

119896=1

infin

sum

119897=0

(minus1)119896+119897

1198882119896+12119897+2

(119892) 1198692119896(119905) 119869

2119897+1(1199051015840

)]

(28)

the use of the property valid for 119896 gt 0

int

+infin

minusinfin

119889119905120601119867(119896 119905) 119891 (119905)

= 81198941205872

1198922

int

+infin

0

119889119905119890minus(119905+119896)2

(radic2119892119896radic2119892119905) 119891 (119905)

119891 (119905) = 119891 (minus119905)

(29)

and the restriction to 119896 gt 0 allow to write the equation for119878(119896) in the alternative way

119878 (119896) =

119871

119896

(1 minus 1198690(radic2119892119896))

minus 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

sdot [

120587119905

sinh (1199052)119878 (119905) minus

2119894119905

1 minus 119890minus119905 (119905) +

119894119905

1 minus 119890minus119905Φ0(119905)

+ (

119894119905

1 minus 119890minus119905

1206010(119905) + 2120587)

119871

sum

ℎ=1

119890119894119905119909ℎ]

(30)

Equations (30) and (26) are our starting points for study-ing ABA contributions to anomalous dimension of twistoperators As planned in the Introduction we will considerthe minimal anomalous dimension state go to the highspin limit and determine the predictions of ABA for theanomalous dimension up to orders 1119904(ln 119904)119899 119899 ge minus1 Wetherefore discuss in next section all the simplifications that(30) undergoes in the high spin limit

3 Ground State and High Spin Limit

In this section we start our study of the minimal anomalousdimension state For this state the positions of the internalholes are as close as possible to the origin that is they satisfythe relations

119885 (119909ℎ) = 120587 (2ℎ + 1 minus 119871) ℎ = 1 119871 minus 2 (31)

while the positions of the two external holes are determinedafter solving the equations

119885 (119909119871minus1) = minus119885 (119909

119871) = 120587 (119904 + 119871 minus 1) (32)

It follows that the positions of the Bethe roots 119906119897are all greater

in modulus than the positions of the internal holes that is|119906119897| gt 119909

ℎ ℎ = 1 119871 minus 2 For our convenience we order

Bethe roots 119906119897in such a way that 119906

119897lt 119906

1198971015840 if 119897 lt 1198971015840

In the following we will find useful to integrate overthe region in which Bethe roots are contained It is thenvery important to make the most convenient choice for theldquoextremardquo of integration which naturally identify the pointsplusmn119887which separate the lastfirst root 119906

119904119906

1(119885(119906

119904119906

1) = ∓120587(119904+

119871 minus 3)) from the positivenegative external hole 119909119871119909

119871minus1 we

choose 119887 such that

119885 (plusmn119887) = ∓120587 (119904 + 119871 minus 2) (33)

Then we perform our analysis of the minimal anomalousdimension state in the high spin limitWe have to remark thatin this limit the set of operators (1) has been the object of anextensive activity [10 39 46ndash52 55ndash67] also in perturbativeQCD see [68ndash73] In the high spin limit the position of theinternal holes is proportional to 1 ln 119904 so it is very close to theorigin they will be determined by using (31) in Section 4 Onthe other hand in order to estimate the position of the two


external or ldquolargerdquo holes we have to evaluate the countingfunction 119885(119906) near the points plusmn119887 119887 sim 119904 delimiting theinterval in which Bethe roots reside The result we will findat their leading orders 119874(119904) and 119874(1199040)

119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)

is proved in next subsection We have to mention that thesame formula (34) was found for twist two in [65] by usingresults of [74 75] However as far as we have understoodresults of [74 75] are proved only at one and two loopsTherefore we would like to give a different and more generalproof of (34)

31 Position of the External Holes When the spin is largeBethe roots near the two ldquoextremardquo plusmn119887 scale with 119904 In theproximity of plusmn119887 it is therefore convenient to rescale thevariable 119906 of the counting function119885(119906) we will write 119906 = 119906119904where 119906 will stay finite Analogously we will define 119887 = 119887119904with 119887finite From the definitions (14) and (16) of the countingfunction we have

119885 (119906119904) = minus2119871 arctan 2119906119904 minus 2119904

sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)

We observe that the only ldquohigher loopsrdquo effect is in the lastterm proportional to the anomalous dimension For 119906 =

119906119897 where 119906

119897119904 = 119906

119897is a Bethe root we expand the various

functions for large 119904 and evaluate the sumover the Bethe rootscontained in (35) as an integral term plus an ldquoanomalyrdquo [76ndash79] We obtain

119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)

and where we used the relation [76ndash79]

minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)

We remark that in order to obtain the last equality in (38) itis crucial to transform the sum into an integration from minus119887 to119887 where 119887 = 119887119904 satisfies (33) If for instance we transformthe sum over Bethe roots and holes into an integration fromthe first 119906

1= minus119906

119904to the last 119906

119904root we obtain an extra119874(1119904)

term 1(119906119897minus119906

119904)+1(119906

119897+119906

119904) in the last line of (38) specifically

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)

Sticking to formula (38) we remember that for the minimalanomalous dimension state and with our ordering of Betheroots the value of the counting function on a generic root 119906

119897

is given by the simple formula

119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)

Property (40) allows to simplify equation (36) as follows

0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)

At the leading order 119874(1199040) we know that the equation to besatisfied for all 119906 is

0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)


whose solution is the well-known [55 80] density

120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)

Using (43) we give an estimate of the last term in (41)

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)

which allows to find the function 120588(119906) which satisfies (41)

120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)

Using the form (45) of the solution we can determine theposition of the extremum 119887 through the relation

int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)

where we used (33) which gives

119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)

We remark that if we had transformed the sum over Betheroots and holes into an integration from 119906

1= minus119906

119904to 119906

119904

according to (39) by repeating all the steps until (47) wewould have found the position of the largest root at leadingand subleading order this result is

119906119904= 119887 + 119874(

1

119904

) (48)

which in particular allows to give an estimate for 1198851015840

(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)

We will use this result for 1198851015840

(119887) in next subsection

We now pass to determine the position 119909119871= 119909

119871119904 119909

119871gt 119887

of the positive external hole We first compute (35) for |119906| gt|119887| (more precisely |119906| minus |119887| = 119874(1))


sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)

The sum over the Bethe roots is evaluated as

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus 2

119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)

We now insert (45) into (51) and use the result valid for |119906| gt119887

int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)

Inserting the resulting expression for (51) into (50) weeventually arrive at the formula

119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)


which is certainly valid for |119906| minus |119887| = 119874(1) Now the positionof the (positive) external holes is fixed by the condition119885(119909

119871119904) = minus(119871 + 119904)120587 + 120587 Imposing that on (53) we find

119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)

We observe that such result agrees when119871 = 2with the zeroesof the transfer matrix which one can obtain from expressionscontained in [74 75] This is an important check for ourfindings

32 High Spin Limit of Nonlinear Terms Another importantsimplification occurring for large spin concerns the nonlinearterm (containing 1015840(119905)) which appears in (30) In this subsec-tion we extend to all loops the result of [47ndash52] Some of theresults of this section have been already announced but notcompletely proved in [81] We will fill that gap here at theend we are able to show that

1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)

Thismeans that in our approximation nonlinearity effects in(30) are under control

In our equation (30) nonlinearity appears in the followingintegral

119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)

It is convenient to pass to the coordinate space and to define

119868120572

(119906) = minus2int

+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840

(120572 minus 119894119906 + 119894V) minus 1205951015840 (120572 + 119894119906 minus 119894V)] 119871 (V)

(57)

We can keep 120572 generic having in mind that the case 120572 = 12is relevant for our case (56)

119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906

sdot cos 119905V (radic2119892119896radic2119892119905)

(59)

In general we split 119868120572(119906) as 119868120572(119906) = 119868120572in(119906) + 119868120572

out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840

(120572 minus 119894119906 + 119894V) minus 1205951015840 (120572 + 119894119906 minus 119894V)]

sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)

Then 119868120572in(119906) is evaluated using formula (217) of [82]

119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)

Now we remember that 1198851015840

(119887) = 119874(119887) (see (49)) in addi-tion in the high spin limit we are allowed to consider119906 ≪ 119904 Therefore we conclude that 119868120572in(119906) = 119874(1119904

2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)

Since we can restrict 119868120572(119906) to |119906| ≪ 119904 we develop the 120595functions in the integrand for large V We obtain

119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)

Integrating by parts we can write down

119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)

We then use the fact that 119871(119887) = 0 and the identity

ln119909119871= minusint

+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)

In order to perform the integration in (66) we need anestimate of 1198851015840

(V) when V gt 119887 In Appendix B we prove that

1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)

Integration in (66) is then performed exactly

minus

8119887

120587

int

+infin

119887

119889Vln VV

1

radicV2 minus 1198872= minus4 ln 119887 minus 4 ln 2 (68)

Plugging (68) into (66) and using the equality 119909119871= radic2119887 +

119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)

Passing now to the kernel 119870 its evaluation in the coordinatespace shows the following behaviour

119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922

119890minus2119894 arcsin(119906radic2119892)

2119909 (V)2])

|119906| le radic2119892 |V| ge radic2119892

(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)

whereΛ sim 119904 is a cutoff such that for119906 lt Λ approximation (69)can be used When 119904 rarr +infin in the first integral we replaceΛ with +infin in the second integral we estimate 119870 using (70)and 11986812(V) by means of (57) using that 1205951015840(119911) sim 1119911 for large119911

V 997888rarr +infin 997904rArr

119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)

which therefore imply that

Λ 997888rarr infin 997904rArr

int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)

Putting all together we find out that

119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)

4 High Spin Results from ABAUp to Order 1119904

Having analysed all the simplifications occurring in the highspin limit let us come back to (30) We insert formula (34)for the position of the external holes use relation (55) for the


nonlinear term and work out all the ldquoknownrdquo terms We endup with the following integral equation

119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1

(ln 119904)minusinfin)

(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840

) minus (119905 0) The particular form ofthe known terms in (75) together with condition (31) for theinternal holes suggests that 119878(119896) expands in (inverse) powers(with 119874(119904

minus1

(ln 119904)minusinfin) we denote terms going to zero fasterthan 1119904 times any inverse power of ln 119904) of ln 119904

119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)

And consistently with (76) the condition for the internalholes (31) is solved by the following Ansatz on their positions

119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904

) (ln 119904)minus119899 + 119874 (119904minus1 (ln 119904)minusinfin) (77)

For the anomalous dimension 120574(119892 119871 119904) = 2119878(0) thereforewe have the expansion

120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)

where the scaling functions 119891(119892) 119891sl(119892 119871) appear also inother contexts for instance119891(119892) is twice the cusp anomalousdimension of Wilson loops [83 84]

For our purposes it is important to remark that the strongcoupling limits of 119891(119892) [63 64] and 119891sl(119892 119871) [65 66] agreewith string theory computations This shows that such func-tions are actually wrapping independent and consequently

the anomalous dimension is wrapping independent at itsleading orders ln 119904 and (ln 119904)0

After these first considerations we come back to (75) theldquoknownrdquo terms driving the equations for 119878(minus1)(119896) and 119878(0)(119896)are contained in the first two lines of (75) The structure ofsuch driving terms implies the following equalities betweendensities

119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)

which translate in terms of anomalous dimensions to theequalities [65 85]

120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)

It is possible to obtain analogous relations for 120574(119899)

(119892 119871)

expressed in terms of the 120574(119899)(119892 119871) for 119899 ge 1 The first step isa standard procedure for integral equations with a separablekernel called Neumann expansion [86] applied to (75) for119878(119896)

119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)

This procedure is fully explained in this application inAppendix C For 119899 = 1 2 3 4 5 we obtain

119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)

119901(119892) belong to a set of ldquoreduced

coefficientsrdquo satisfying the system (C6) reported also inAppendix C

These expressions are still quite involved but they canbe significantly simplified This can be done through thefollowing steps

(i) After introducing the notation

119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)

we ldquoinvertrdquo relation (31) expressing 120572119898ℎ

and 119898ℎ

interms of the densities and their derivatives in zerothat is in terms of the coefficients 120590(119899)

119903and (119899)

119903 In

performing this procedure we use techniques andresults of [87]Then we plug the obtained expressionsfor 120572

119898ℎ

119898ℎin (82) Detailed calculations are shown

in Appendix D where we have also listed the fullexpressions for the first 119878

(119899)

119901(119892 119871) (relations (D4))

(ii) Then we use the following relations proven inAppendix E

119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0

minus (119871 minus 1)

120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)

With the help of these formulae it is possible to comparethe complicated relations (D4) with analogous results for

119878(119899)

119901(119892 119871) found in [87] and reported in Appendix F ending

up with the following simple and compact expressions

119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)

For anomalous dimensions such relations read

120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)

Relations (80) and (86) are the prediction of ABA foranomalous dimensions at various orders 1119904(ln 119904)119899 119899 =

minus1 5 In the next section we will show that they agreewith predictions coming from self-tuning and reciprocityrelations


5 1119904 Contributions fromFunctional Relations

Self-tuning and reciprocity relations were summarised informulae (3) (4) and (5) We remember notations (78) forthe high spin expansion of the anomalous dimension

120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)

Comparing (87) with (3) (4) and (5) we obtain that theleading terms of 119875(119904) should read

119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)

Developing 119862(119904) in the same regime

119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)

and putting together (3) (88) and (89) we end up with thefollowing prediction for the anomalous dimension

120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)

Working out this formula for orders 1119904(ln 119904)119899 119899 = minus1 5we find formulae which coincide with (80) for 119899 = minus1 0 andwith (86) for119899 = 1 5Therefore our findings in Section 4agree with self-tuning and reciprocity predictions We wouldlike to stress the fact that this conclusion holds for all values ofthe coupling119892 that is it is a nonperturbative statement on thehigh spin expansion of (asymptotic) anomalous dimension

Remark 1 Formulae (79) and (85) seem to indicate that moregenerally functional relations similar to (3) and (4) shouldhold for (the high spin expansion of) the function 119878(119896) On

the basis of our results we are naturally led to make thefollowing proposal for a self-tuning relation involving thecoefficients of the Neumann expansion of the function 119878(119896)

S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)

whereP119901(119904) satisfies a high spin expansion analogous to (4)

P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)

where 119862(119904) is given by (5) In particular formula (91) hasthe advantage to furnish immediately the self-tuning (andreciprocity) relations for all the higher conserved charges[88]

119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)

Furthermore we may suppose that because of integrabilitythe statement above is equivalent to the self-tuning (andreciprocity) of the counting function (91)

Remark 2 Concerning the leading terms in the high spinexpansion (78) of the minimal anomalous dimension wealready commented that since the strong coupling limit of119891(119892) and 119891sl(119892 119871) agrees with string theory calculations (andmany gauge loop calculations) there are good reasons tobelieve that anomalous dimension at the orders ln 119904 and(ln 119904)0 is free from wrapping contributions Then if we sup-pose that self-tuning and reciprocity are exact symmetriesfrom (80) and (90) it follows that also 120574

(minus1) and 120574(0) are

wrapping-free Then one could expect that all the terms thatare in between 119891sl(119892 119871) and 120574

(minus1) (the various 120574(119899)(119892 119871)) arealso not affected by wrapping If we suppose this use againself-tuning and reciprocity and compare (86) with (90) weare able to conclude that all the functions 120574

(119899)

(119892 119871) do notdepend on wrapping either

Even if we are aware that our arguments do not providea proof of the fact that at high spin wrapping diagrams startcontributing at orders (ln 119904)21199042 we however think that ourresults provide some nonperturbative hints of this property

Remark 3 In the previous remark we gave some evidence infavour of the fact that when 119904 rarr +infin and the twist 119871 is fixedfor all values of the coupling constant wrapping diagramsstart contributing at the order (ln 119904)21199042 We would like tostress that this conclusion depends on the particular order inwhich limits are performed indeed we first sent 119904 rarr +infinwith 119871 and 119892 fixed then possibly we could have made thelimit 119892 rarr +infin Obviously our situation is different fromwhat happens in the peculiar regime of semiclassical stringsin this case indeed 119904 and 119892 go together to infinity with theirratio S = 1199042radic2120587119892 kept fixed and wrapping contributionenters already at the order 1 lnS [89ndash92]

6 Conclusions

We studied the high spin limit of twist operators in the sl(2)sector ofN = 4 SYM Using ABA equations rewritten as one


NLIE we computed the minimal anomalous dimension upto orders 1119904(ln 119904)119899mdashin detail Section 4 and formulae (86)mdashproving eventually that our results satisfy (for all values ofthe coupling 119892) the self-tuning and reciprocity properties (3)(4) and (5) As a consequence in Remark 2 above we couldgive some clues supporting the idea that in the high spin limitwrapping corrections start contributing at order (ln 119904)21199042 forany twist 119871

In addition as a byproduct of our analysis we providedalso the following new results

(1) Exact connection between a nonlinear function of thecounting function and the (asymptotic) anomalousdimension (formula (26) and Appendix A)

(2) Evaluation of the external holes position at the (sub-leading) order 1199040 (Section 31)

(3) Proposal for self-tuning and reciprocity relations sat-isfied by the function (directly related to the countingfunction) 119878(119896) (Remark 1 of Section 5 and formulae(91) and (92)

Eventually we are confident that further analysis of thelast issue may shed light on how to construct order by orderin 119904 an ldquoeffectiverdquo (non)linear integral equation which takesinto account the wrapping corrections as well

Appendices

A Connection between the Counting Functionand (Asymptotic) Anomalous Dimension

It is convenient to rewrite (27) in the following form

119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)

Then we compute (A1) at 119896 = 0 We obtain

119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)

is the Fourier transform of

119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)

On the other hand if we want to compute the anomalousdimension 120574(119892 119871 119904) of a state described by a solution ofthe ABA equations we have to compute the sum 120574 =

1198922

sum119904

119896=1119890(119906

119896) Using formula (19) we obtain

120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)

which can be written also in terms of Fourier transforms as

120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)

Comparing (A6) with (A2) we gain relation (26)For an alternative proof of (26) we first notice that

(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)

Then using (18) we find that when 119906 rarr plusmninfin

119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)

Using these property one finds that

lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)

Inserting (A7) and (A9) into (25) we find again (26) Thisalternative proof emphasizes the fact that the information onthe anomalous dimension comes entirely from the term (119896)which is a nonlinear function of the counting function 119885(119906)

lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin

119889119906119890minus119894119896119906 Im ln [1 + (minus1)119871 119890119894119885(119906minus1198940

+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)

Curiously (and perhaps interestingly) formula (A10) lookssimilar to the TBA expression for the free energy We havechecked formula (A10) in various particular cases (twist 2 3one and two loops) for which explicit solutions [93] of ABAequations were found1

B Evaluation of 1198851015840

(119906) at High Spinand Large 119906

Using definition (14) we write the counting function 119885(119906) inthe region of large 119906 sim 119887

119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)


We add and subtract the sum over the internal holes and thusobtain

119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1

arctan (119906 minus 119909ℎ) + 120587 (119871 minus 2) minus

2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)

Then the use of (19) gives

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)

Evaluation of the nonlinear term is done using formula (217)

of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V)

sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1

arctan (119906 minus 119885(minus1)

(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)

where we neglected higher order terms in the sum over 119896since 1198851015840

(119887) sim 119874(119887) (49) It is convenient to pass to the deriv-ative of 119885

1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)

For large 119906 but still 0 lt 119906 lt 119887 the solution to (B5) is

1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2

119887 minus radic1198872minus 119906

2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)

Indeed if we insert (B6) into the integral of (B5) and use theintegration formula

int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2

119887 minus radic1198872minus V2

= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887

radic1198872minus 119911

2+ 119887

119911 notin [minus119887 119887]

(B7)

in the right-hand side of this last equation (with 119911 = 119906 plusmn 119894)when 0 lt 119906 lt 119887 we are left with

minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887

radic1198872minus (119906 minus 119894)

2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)

whichmatches (B6) Plugging approximation (B6) into (B5)and letting 119906 gt 119887 we find 1198851015840

(119906) in this domain Applicationof (B7) gives

1198851015840

(119906) = minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)

which since 119906 gt 119887 ≫ 1 is expanded as follows

1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)

C Neumann Expansion for 119878(119896)

Let one see how the Neumann expansion for 119878(119896)worksThisis a standard procedure [86] in the case of an integral equationwith separable kernel


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]

sdot (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)

TheNeumann expansion transforms the linear integral equa-tion for 119878(119896) into a set of linear infinite system In particular119878(119899)

119901(119892 119871) 119899 ge 1 satisfy the system2

119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)

minus (2119901 minus 1)int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)

minus 2 (2119901 minus 1)

infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)

where (119899)(119892 119905) appears in the expansion

119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)

which follows from (77) More explicitly inserting (77) into(C3) we obtain

(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)

Plugging (C4) into (C2) we obtain (119899 ge 1)

119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)

where the ldquoreduced coefficientsrdquo 119878(119903)119901(119892) satisfy the systems

(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)

D Solution of Internal Holes Equation 119899ℎ

Expressed in Terms of the Densities andTheir Derivatives

We want to extract from (31) explicit expressions for 120572119898ℎ

and 119898ℎ

in terms of the densities and their derivativesusing the notation (83) We can use techniques and resultsof [87] In particular relations (14) of [87] are still valid aftersubstituting in them

120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)

We can then solve for 119898ℎ

We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)

119903minus1(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)

where the coefficients 119895119898of the first term in the right-hand

side satisfy sum119901minus119903+1

119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the

coefficients 119895119898related to the second term satisfysum119901minus119903minus119897+1

119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)

Now inserting (D3) into (82) we can derive expressions for119878(119899)

119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)

3(119871 minus 1) (119871 minus 2) (119871 minus 3)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871

minus 2) (119871 minus 3)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]

(119871 minus 1) (119871 minus 2) (119871 minus 3)

(D4)


E Useful Relations

It is possible to express certain ratios among coefficients 120590(119899)119903

and (119899)119903

in terms of functions 119891(119892) 119891sl(119892 119871) and 120574(119899)

(119892 119871)We nowfind some of these relations which are useful in orderto prove (85)

Let us start with (214) of [87] and (C4) Comparingthem we find the following relation

(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)

Now developing 119878(119896) according to relation (76)

119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)

and using integral equation (75) together with (78) it ispossible to obtain the following relations

119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)

For what concerns (119896) we have the exact expression

(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

(cos 119896119909ℎminus 1) minus

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)

Then applying inverse Fourier transform we obtain

120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]

minus (119871 minus 2) [120595 (1 minus 119894119906) + 120595 (1 + 119894119906)]

minus 120595 (1 minus 119894119909119871minus 119894119906) minus 120595 (1 + 119894119909

119871minus 119894119906)

minus 120595 (1 minus 119894119909119871+ 119894119906) minus 120595 (1 + 119894119909

119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)

Using the position of the external holes (34) and computing120590(119906) at 119906 = 0 we obtain

120590 (0) = minus4 ln 119904 minus 4120574119864minus 4119871 ln 2 + 119866 (0) minus 2119891 (119892) ln 119904

119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)

sdot (ln 119904)minus119899 + 119874 (119904minus1 (ln 119904)minusinfin)

(E7)

It is obvious that expanding 119866(0) in the same way of 119878(119896)in (E2) relations (E3) are also valid for the correspondingcoefficients of 119866(0) Using these relations and also (E1) itis possible to find from (E7) and remembering (83) thefollowing relations

119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)

Computing from (E6) the second derivative of 120590(119906) at 119906 = 0it is also possible to show that

119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)

F Explicit Expressions for 119878(119899)119901(119892119871) with

119899 = 1 2 3 4 5

We report here the expressions of the functions 119878(119899)119901(119892 119871)

with 119899 = 1 2 3 4 5 in terms of the densities (and theirderivatives in zero) and the solutions of the ldquoreduced systemsrdquo119878(119899)

119901(119892) The general method to obtain them and results for

119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)

2(119871 minus 3) (119871 minus 2) (119871 minus 1) 119878

(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)

119901(119892 119871) = 2120587 (119871 minus 3) (119871 minus 2) (119871 minus 1)

[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)

119901(119892 119871) = (119871 minus 3) (119871 minus 2) (119871 minus 1)

sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank gratefully B Basso for scientific discus-sions As for travel financial support INFN IS Grant GASTtheUniTo-SanPaolo researchGrant no TO-Call3-2012-0088the ESF Network HoloGrav (09-RNP-092 (PESC)) andthe MPNS-COST Action MP1210 are kindly acknowledgedGabriele Infusino acknowledges EU Italian Republic andCalabria Region for funding through Regional OperativeProgram (ROP) Calabria ESF 20072013mdashIV Axis HumanCapital-Operative Objective M2

Endnotes

1 Results in thisAppendix could seem in contrastwithBESfinding stating that anomalous dimension at order ln 119904 isgiven by the value in zero of the Fourier transform of theldquohigher than one loop density of rootsrdquo which satisfiesBES equation The solution to this apparent contrast is

that the function Σ(119896) which satisfies the BES equationsuch that Σ(0+) = 120587120574(119892 119871 119904)|ln 119904 in our notations reads

Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)

where the label119867means that only higher than one loopcontributions have to be included Then in addition towhat we would call ldquohigher than one loop density ofrootsrdquo that is

119867(119896) in BES density (lowast) there is also

a term nonlinear in the counting function 119885 This termcan be estimated at large 119904 by using (69) to be (almosteverywhere)119874(11199042) therefore as far as the order ln 119904 isconcerned it is almost everywhere negligible with theexception of the point 119896 = 0 where one experiencesthe noncommutativity between the limits 119904 rarr +infin and119896 rarr 0 However the nonlinear term has to be kept indefinition (lowast) of the BES density since it gives the entireinformation on anomalous dimensions (see (A10)) dueto the fact that

119867(0) = (0) minus |

1loop(0) = 02 We use the notation (119869

119899is a Bessel function)

119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References

[1] J M Maldacena ldquoThe large N Limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[2] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters BNuclear Elementary Particle and High-Energy Physics vol 428no 1-2 pp 105ndash114 1998

[3] E Witten ldquoAnti de sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 pp 253ndash291 1998

[4] J Minahan and K Zarembo ldquoThe bethe-ansatz for N = 4 superyang-millsrdquo Journal of High Energy Physics vol 2003 article 032003

[5] N Beisert andM Staudacher ldquoTheN = 4 SYM integrable superspin chainrdquoNuclear Physics B vol 670 no 3 pp 439ndash463 2003

[6] V Kazakov A Marshakov J Minahan and K ZaremboldquoClassicalquantum integrability in AdSCFTrdquo Journal of HighEnergy Physics vol 2004 no 5 article 024 2004

[7] M Staudacher ldquoThe factorized S-matrix of CFTAdSrdquo Journalof High Energy Physics vol 2005 no 5 article 054 2005

[8] N Beisert V A Kazakov K Sakai and K Zarembo ldquoTheAlgebraic curve of classical superstrings on119860119889119904

5times119878

5rdquoCommu-nications in Mathematical Physics vol 263 no 3 pp 659ndash7102006

[9] N Beisert and M Staudacher ldquoLong-range psu(22mdash4) Betheansatze for gauge theory and stringsrdquoNuclear Physics B vol 727no 1-2 pp 1ndash62 2005

[10] N Beisert B Eden andM Staudacher ldquoTranscendentality andcrossingrdquo Journal of StatisticalMechanics vol 2007 no 1 ArticleID P01021 2007

[11] J Ambjoslashrn R A Janik and C Kristjansen ldquoWrapping interac-tions and a new source of corrections to the spin-chainstringdualityrdquo Nuclear Physics B vol 736 no 3 pp 288ndash301 2006


[12] A V Kotikov L N Lipatov A Rej M Staudacher and VN Velizhanin ldquoDressing and wrappingrdquo Journal of StatisticalMechanics vol 10 Article ID P10003 2007

[13] D Bombardelli D Fioravanti and R Tateo ldquoThermodynamicBethe ansatz for planar AdSCFT a proposalrdquo Journal of PhysicsA Mathematical and Theoretical vol 42 no 37 Article ID375401 2009

[14] N Gromov V Kazakov A Kozak P Vieira and N GromovldquoExact spectrum of anomalous dimensions of planar N = 4supersymmetric Yang-Mills theory TBA and excited statesrdquoLetters in Mathematical Physics vol 91 no 3 pp 265ndash287 2010

[15] G Arutyunov and S Frolov ldquoThermodynamic bethe ansatz forthe119860119889119904

5times119878

5 mirrormodelrdquo Journal of High Energy Physics vol2009 no 5 article 068 2009

[16] D Bombardelli D Fioravanti and R Tateo ldquoTBA andY-systemfor planar AdS

4CFT

3rdquo Nuclear Physics B vol 834 no 3 pp

543ndash561 2010[17] N Gromov and F Levkovich-Maslyuk ldquoY-system TBA and

Quasi-Classical strings in AdS4x CP3rdquo Journal of High Energy

Physics vol 2010 article 088 2010[18] N Gromov V Kazakov and P Vieira ldquoExact spectrum of

anomalous dimensions of planar N=4 supersymmetric Yang-Mills theoryrdquo Physical Review Letters vol 103 no 13 Article ID131601 2009

[19] A Cavaglia D Fioravanti andR Tateo ldquoExtendedY-system fortheAdS

5CFT

4correspondencerdquoNuclear Physics B vol 843 no

1 pp 302ndash343 2011[20] S S Gubser I R Klebanov and A M Polyakov ldquoA semi-

classical limit of the gaugestring correspondencerdquo NuclearPhysics B vol 636 no 3 pp 99ndash114 2002

[21] S Frolov andA Tseytlin ldquoSemiclassical quantization of rotatingsuperstring in AdS

5times S5rdquo Journal of High Energy Physics vol

2002 no 6 article 007 2002[22] D Fioravanti S Piscaglia and M Rossi ldquoOn the scattering

over the GKP vacuumrdquo Physics Letters Section B NuclearElementary Particle and High-Energy Physics vol 728 no 1 pp288ndash295 2014

[23] D Fioravanti S Piscaglia and M Rossi ldquoAsymptotic BetheAnsatz on the GKP vacuum as a defect spin chain scatteringparticles andminimal areaWilson loopsrdquoNuclear Physics B vol898 pp 301ndash400 2015

[24] L N Lipatov ldquoReggeization of the vector meson and the vac-uum singularity in nonabelian Gauge theoriesrdquo Soviet Journalof Nuclear Physics vol 23 pp 338ndash345 1976 Yadernaya Fizikavol 23 pp 642ndash656 1976

[25] S Moch J A M Vermaseren and A Vogt ldquoThe three-loopsplitting functions in QCD the non-singlet caserdquo NuclearPhysics B vol 688 no 1-2 pp 101ndash134 2004

[26] A Vogt S-O Moch and J A M Vermaseren ldquoThe three-loopsplitting functions in QCD the singlet caserdquo Nuclear Physics Bvol 691 no 1-2 pp 129ndash181 2004

[27] Y L Dokshitzer G Marchesini and G P Salam ldquoRevisitingparton evolution and the large-x limitrdquo Physics Letters B vol634 no 5-6 pp 504ndash507 2006

[28] B Basso and G P Korchemsky ldquoAnomalous dimensions ofhigh-spin operators beyond the leading orderrdquo Nuclear PhysicsB vol 775 no 1-2 pp 1ndash30 2007

[29] Y L Dokshitzer G Marchesini and G P Salam ldquoN = 4 SUSYYang-Mills three loops made simple(r)rdquo Physics Letters B vol646 no 4 pp 189ndash201 2007

[30] M Beccaria Y L Dokshitzer andGMarchesini ldquoTwist 3 of thesl(2) sector of N=4 SYM and reciprocity respecting evolutionrdquoPhysics Letters B vol 652 no 4 pp 194ndash202 2007

[31] M Beccaria and V Forini ldquoReciprocity of gauge operators inN = 4 SYMrdquo Journal of High Energy Physics vol 2008 no 6article 077 2008

[32] M Beccaria and V Forini ldquoQcd-like properties of anomalousdimensions in the 119873 = 4 supersymmetric Yang-Mills theoryrdquoTheoretical and Mathematical Physics vol 159 no 3 pp 712ndash720 2009

[33] M Beccaria and V Forini ldquoFour loop reciprocity of twist twooperators in N = 4 SYMrdquo Journal of High Energy Physics vol2009 no 3 article 111 2009

[34] M Beccaria and G Macorini ldquoReciprocity and integrability inthe sl(2) sector of N = 4 SYMrdquo Journal of High Energy Physicsvol 2010 no 1 article 031 2010

[35] M Beccaria V Forini and G Macorini ldquoGeneralized Gribov-Lipatov reciprocity and AdSCFTrdquo Advances in High EnergyPhysics vol 2010 Article ID 753248 37 pages 2010

[36] G Georgiou andG Savvidy ldquoLarge spin behavior of anomalousdimensions and short long strings dualityrdquo Journal of Physics AMathematical andTheoretical vol 44 no 30 Article ID 3054022011

[37] M Beccaria G Macorini and C A Ratti ldquoWrapping correc-tions and reciprocity beyond the sl(2) subsector inN = 4 SYMrdquoJournal of High Energy Physics vol 2011 no 6 article 071 2011

[38] L Alday A Bissi and T Lukowski ldquoLarge spin systematicsinCFTrdquo httparxivorgabs150207707

[39] D Bombardelli D Fioravanti and M Rossi ldquoLarge spincorrections in N = 4 SYM 119904119897(2) still a linear integral equatiordquoNuclear Physics B vol 810 no 3 pp 460ndash490 2009

[40] Z Bajnok R A Janik and T Łukowski ldquoFour loops twist twoBFKL wrapping and stringsrdquo Nuclear Physics B vol 816 no 3pp 376ndash398 2009

[41] T Łukowski A Rej and V Velizhanin ldquoFive-loop anomalousdimension of twist-two operatorsrdquo Nuclear Physics B vol 831no 1-2 pp 105ndash132 2010

[42] V Velizhanin ldquoSix-loop anomalous dimension of twist-threeoperators in N = 4 SYMrdquo Journal of High Energy Physics vol2010 no 11 article 129 2010

[43] G Arutyunov S Frolov and M Staudacher ldquoBethe ansatz forquantum stringsrdquo Journal of High Energy Physics vol 2004 no10 article 016 2004

[44] N Beisert R Hernandez and E Lopez ldquoA crossingndashsymmetricphase for119860119889119878

5times119878

5 stringsrdquo Journal of High Energy Physics vol2006 article 11 Article ID 0609044 2006

[45] C Destri and H J de Vega ldquoUnified approach to Thermody-namic BetheAnsatz and finite size corrections for latticemodelsand field theoriesrdquo Nuclear Physics B vol 438 no 3 pp 413ndash454 1995

[46] A V Belitsky A S Gorsky andG P Korchemsky ldquoLogarithmicscaling in gaugestring correspondencerdquo Nuclear Physics B vol748 no 1-2 pp 24ndash59 2006

[47] L Freyhult A Rej and M Staudacher ldquoA generalized scalingfunction for AdSCFTrdquo Journal of Statistical Mechanics Theoryand Experiment vol 2008 no 7 article P015 2008

[48] A V Belitsky G P Korchemsky and R S Pasechnik ldquoFinestructure of anomalous dimensions in N=4 super-Yang-Millstheoryrdquo Nuclear Physics B vol 809 no 1-2 pp 244ndash278 2009

[49] D Fioravanti P Grinza and M Rossi ldquoStrong coupling forplanarN = 4 SYM theory an all-order resultrdquo Nuclear PhysicsB vol 810 no 3 pp 563ndash574 2009


[50] B Basso and G P Korchemsky ldquoEmbedding nonlinear O(6)sigma model into N= 4 super-Yang-Mills theoryrdquo NuclearPhysics B vol 807 no 3 pp 397ndash423 2009

[51] D Fioravanti P Grinza and M Rossi ldquoThe generalised scalingfunction a noterdquoNuclear Physics B vol 827 no 3 pp 359ndash3802010

[52] D Fioravanti P Grinza and M Rossi ldquoThe generalised scalingfunction a systematic studyrdquo Journal of High Energy Physics vol2009 no 11 article 037 2009

[53] D Fioravanti A Mariottini E Quattrini and F RavaninildquoExcited state Destri-De Vega equation for sine-Gordon andrestricted sine-Gordon modelsrdquo Physics Letters B vol 390 no1ndash4 pp 243ndash251 1997

[54] D Fioravanti and M Rossi ldquoFrom finite geometry exactquantities to (elliptic) scattering amplitudes for spin chains the12-XYZrdquo Journal of High Energy Physics vol 2005 no 8 article010 2005

[55] B Eden and M Staudacher ldquoIntegrability and transcendental-ityrdquo Journal of StatisticalMechanicsTheory and Experiment vol2006 no 11 Article ID P11014 2006

[56] S Frolov A Tirziu andAA Tseytlin ldquoLogarithmic correctionsto higher twist scaling at strong coupling from AdSCFTrdquoNuclear Physics B vol 766 no 1ndash3 pp 232ndash245 2007

[57] M K Benna S Benvenuti I R Klebanov and A ScardicchioldquoTest of the anti-de sitter-spaceconformal-field-theory corre-spondence using high-spin operatorsrdquo Physical Review Lettersvol 98 no 13 Article ID 131603 2007

[58] L F Alday G Arutyunov B Eden M K Benna and I RKlebanov ldquoOn the strong coupling scaling dimension of highspin operatorsrdquo Journal of High Energy Physics vol 2007 no 4article 082 2007

[59] I Kostov D Serban and D Volin ldquoStrong coupling limit ofBethe ansatz equationsrdquo Nuclear Physics B vol 789 no 3 pp413ndash451 2008

[60] M Beccaria G F De Angelis and V Forini ldquoThe scalingfunction at strong coupling from the quantum string Betheequationsrdquo Journal of High Energy Physics vol 2007 article 42007

[61] P Y Casteill and C Kristjansen ldquoThe strong coupling limit ofthe scaling function from the quantum string Bethe ansatzrdquoNuclear Physics B vol 785 no 1-2 pp 1ndash18 2007

[62] N Gromov ldquoGeneralized scaling function at strong couplingrdquoJournal ofHigh Energy Physics vol 2008 no 11 article 085 2008

[63] B Basso G P Korchemsky and J Kotanski ldquoCusp anomalousdimension in maximally supersymmetric yang-mills theory atstrong couplingrdquo Physical Review Letters vol 100 no 9 ArticleID 091601 2008

[64] I Kostov D Serban and D Volin ldquoFunctional BES equationrdquoJournal of High Energy Physics vol 2008 no 8 article 101 2008

[65] L Freyhult and S Zieme ldquoVirtual scaling function of AdSCFTrdquoPhysical Review D vol 79 no 10 Article ID 105009 2009

[66] D Fioravanti P Grinza and M Rossi ldquoBeyond cusp anoma-lous dimension from integrabilityrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 675no 1 pp 137ndash144 2009

[67] D Fioravanti G Infusino and M Rossi ldquoOn the high spinexpansion in the 119904(2)N=4 SYM theoryrdquoNuclear Physics B vol822 no 3 pp 467ndash492 2009

[68] V M Braun S E Derkachov and A N Manashov ldquoIntegra-bility of three-particle evolution equations in QCDrdquo PhysicalReview Letters vol 81 no 10 pp 2020ndash2023 1998

[69] V M Braun S E Derkachov G P Korchemsky and A NManashov ldquoBaryon distribution amplitudes in QCDrdquo NuclearPhysics B vol 553 no 1-2 pp 355ndash426 1999

[70] A V Belitsky ldquoFine structure of spectrum of twist-threeoperators in QCDrdquo Physics Letters B vol 453 no 1-2 pp 59ndash72 1999

[71] A V Belitsky ldquoIntegrability and WKB solution of twist-threeevolution equationsrdquo Nuclear Physics B vol 558 no 1-2 pp259ndash284 1999

[72] A V Belitsky ldquoRenormalization of twist-three operators andintegrable lattice modelsrdquo Nuclear Physics B vol 574 no 1-2pp 407ndash447 2000

[73] A V Belitsky A S Gorsky and G P KorchemskyldquoGaugestring duality for QCD conformal operatorsrdquo NuclearPhysics B vol 667 no 1-2 pp 3ndash54 2003

[74] A V Belitsky G P Korchemsky and D Muller ldquoTowards Bax-ter equation in supersymmetric Yang-Mills theoriesrdquo NuclearPhysics B vol 768 no 1-2 pp 116ndash134 2007

[75] A V Belitsky ldquoLong-range 119878119871(2) Baxter equation in N = 4

super-Yang-Mills theoryrdquo Physics Letters B vol 643 no 6 pp354ndash361 2006

[76] D V Boulatov ldquoWilson loop on a sphererdquo Modern PhysicsLetters A vol 9 no 4 pp 365ndash374 1994

[77] J M Daul and V Kazakov ldquoWilson loop for largeN Yang-Millstheory on a two-dimensional sphererdquo Physics Letters B vol 335no 3-4 pp 371ndash376 1994

[78] N Beisert A A Tseytlin and K Zarembo ldquoMatching quantumstrings to quantum spins one-loop vs finite-size correctionsrdquoNuclear Physics B vol 715 no 1-2 pp 190ndash210 2005

[79] R Hernandez E Lopez A Perianez and G Sierra ldquoFinite sizeeffects in ferromagnetic spin chains and quantum corrections toclassical stringsrdquo Journal of High Energy Physics vol 2005 no6 article 011 2005

[80] G P Korchemsky ldquoQuasiclassical QCD pomeronrdquo NuclearPhysics B vol 462 no 2-3 pp 333ndash388 1996

[81] D Fioravanti and M Rossi ldquoTBA-like equations and Casimireffect in (non-)perturbative AdSCFTrdquo Journal of High EnergyPhysics vol 2012 no 12 article 013 2012

[82] D Fioravanti and M Rossi ldquoThehigh spin expansion of twistsector dimensions the planar N = 4 super yang-mills theoryrdquoAdvances in High Energy Physics vol 2010 Article ID 61413030 pages 2010

[83] G P Korchemsky ldquoAsymptotics of the altarelli-parisi-lipatovevolution kernels of parton distributionsrdquo Modern PhysicsLetters A vol 04 no 13 pp 1257ndash1276 1989

[84] G P Korchemsky and G Marchesini ldquoStructure function forlarge x and renormalization of Wilson looprdquo Nuclear Physics Bvol 406 no 1-2 pp 225ndash258 1993

[85] M Beccaria V Forini A Tirziu and A A Tseytlin ldquoStructureof large spin expansion of anomalous dimensions at strongcouplingrdquoNuclear Physics B vol 812 no 1-2 pp 144ndash180 2009

[86] A V Kotikov and L N Lipatov ldquoOn the highest transcenden-tality in N = 4 SUSYrdquo Nuclear Physics B vol 769 no 3 pp217ndash255 2007

[87] D Fioravanti P Grinza and M Rossi ldquoOn the logarithmicpowers of 119904119897(2) SYM

4rdquo Physics Letters B vol 684 no 1 pp 52ndash

60 2010[88] G Macorini and M Beccaria ldquoReciprocity ofhigher con-

served charges in the 119904119897(2) sector of N = 4 SYMrdquo httparxivorgabs10095559


[89] M Beccaria G V Dunne V Forini M Pawellek and A ATseytlin ldquoExact computation of one-loop correction to theenergy of folded spinning string in AdS

5times S5rdquo Journal of Physics

A vol 43 no 16 Article ID 165402 2010[90] S Giombi R Ricci R Roiban and A A Tseytlin ldquoTwo-loop

1198601198891198785times 119878

5 superstring testing asymptotic Bethe ansatz andfinite size correctionsrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 4 Article ID 045402 2011

[91] N Gromov D Serban I Shenderovich and D Volin ldquoQuan-tum folded string and integrability from finite size effects toKonishi dimensionrdquo Journal of High Energy Physics vol 2011no 8 article 046 2011

[92] B Basso and A V Belitsky ldquoLuscher formula for GKP stringrdquoNuclear Physics B vol 860 no 1 pp 1ndash86 2012

[93] M Beccaria ldquoAnomalous dimensions at twist-3 in the 119904119897(2)sector ofN = 4 SYMrdquo Journal of High Energy Physics vol 2007no 6 article 044 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


in the maximally supersymmetric theory) In this contexttwo regimes emerge naturally 119909 rarr 0 governed by theBFKL equations [24] and 119909 rarr 1 corresponding exactlyto large values of the Lorentz spin 119904 rarr infin Properties ofthis second (called quasielastic) regime can be deduced bylarge spin results in three-loop twist 2 QCD calculationsIn particular we can highlight two main features aboutanomalous dimension of twist operators

(1) The leading term has a logarithmic scale

120574 (119904) sim ln 119904 119904 997888rarr infin (2)

(2) Subleading terms obey hidden relations the Moch-Vermaseren-Vogt constraints [25 26] in brief termsproportional to ln 119904119904 and 1119904 are completely deter-mined by terms proportional to ln 119904 and 119904

0 Theseconstraints are related with spacetime reciprocity ofdeep inelastic scattering and its crossed version of119890+

119890minus annihilation into hadrons

N = 4 gauge theory shares at large 119904 these features andbesides allows us an understanding of their origin and thuspossible extension to QCD In specific the asymptotic large119904 series of the anomalous dimensions are believed to beconstrained by nonperturbative (in 120582 = 8120587

2

1198922 the rsquot Hooft

multicolor coupling) functional relations that work for anyfinite value of the twist 119871 To be more precise conformalsymmetry implies that anomalous dimensions 120574(119892 119871 119904) oftwist operators are functions of the conformal spin thistranslates into the following ldquoself-tuningrdquo functional relation[27 28]

120574 (119892 119871 119904) = 119875 (119904 +

1

2

120574 (119892 119871 119904)) (3)

Additionally this has to be meant asymptotically in the sensethat the function (the function 119875 in (3) actually depends onthe twist of the operator as well)

119875 (119904) =

infin

sum

119899=0

119886(119899)

(ln119862 (119904))119862 (119904)

2119899 (4)

is represented by a series in 119904 via the conformal Casimir

119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) (5)

only Relation (4) is equivalent to the so-called reciprocitysymmetry 119909 rarr minus119909 but for the Mellin space variable 119904 andan important piece of information is that the function 119886(119899) hasthe form of an upper truncated Laurent series

119886(119899)

(ln119862 (119904)) =119872

sum

119898=minusinfin

119887(119899)

119898(119892 119871) (ln119862 (119904))119898 119872 lt infin (6)

that is 119886(119899)(ln119862(119904)) depends on 119904 only through powers ofln119862(119904) For twist two and three negative powers of ln119862(119904) areabsent and 119886(119899) is a polynomial for generic twist howeverone has to cope with the infinite Laurent series (6)

Relations (3) and (4) both in QCD and in N = 4 SYMare developed checked in various cases and discussed in [27ndash37] Recently they have been proven restrictively to twist twooperators but in a generic conformal field theory in [38]with some arguments for their validity in nonconformal the-ories at the end Clearly they provide important informationon the high spin expansion of anomalous dimension of twistoperators UnlikeQCD inN = 4 SYM it is possible to obtainbetter and more suitable results as we can consider theserelations into the framework of integrability The latter wasfirstly discovered in the planar limit for the purely bosonicso(6) sector at one loop [4] then it was extended to all thegauge theory sectors and to all loops [5ndash10] In specific itwas found that every composite operator can be thoughtof as a state of a ldquospin chainrdquo whose Hamiltonian is thedilatation operator itself although the latter does not havean explicit expression of the spin chain form but for thefirst few loops Nevertheless the spectrum of infinitely longoperators has turned out to be exactly described by a set ofAsymptotic Bethe Ansatz (ABA) equations [5ndash10] On theother hand anomalous dimensions of operators with finitequantum numbers depend not only on ABA data but alsoon finite size ldquowrappingrdquo corrections [11 12] Subsequentprogress has shown that a set of Thermodynamic BetheAnsatz (TBA) equations [13ndash17] or an equivalent 119884-systemof functional equations [18] together with certain additionalinformation [19] provides a solid ground for exact (anylength any coupling) predictions on anomalous dimensionsof planarN = 4 SYM

Despite this impressive progress we believe that it isstill important to define the largest domain of compositeoperators for which the ldquosimplerrdquo ABA equations give thecorrect anomalous dimensions especially in connexion withotherwell-established relevant equations In fact we intendedthis to be the main aim of this paper and the most naturalsetting to perform this study to be the reformulation of ABAequations as one (Non)linear Integral Equation (NLIE) [39]

Generically and sketchily for operators composed of119871 elementary fields ABA gives the correct perturbativeexpansion of the anomalous dimension up to 119871 minus 1 loopsStarting from 119871 loops ldquowrappingrdquo diagrams which are nottaken into account by ABA start to contribute In thisgeneral framework the high spin limit of fixed twist operatorsseems to offer a better scenario Perturbative (up to sixloops) computations [40ndash42] for short (twist two and three)operators show that wrapping diagrams (which enter fromfour loops on) actually give contributions which in the highspin limit behave as 119874((ln 119904)21199042) It is then natural to ask ifsuch property extends to higher (and possibly to all) ordersof perturbation theory In this paper we want to provideevidence in favour of this picture by using the self-tuningand reciprocity properties In order to do that we first rewrite(Section 2) the ABA equations as NLIEs for the countingfunction Then in Section 3 we specialise ourselves to theminimal anomalous dimension state and go to the high spinlimit while keeping the twist finite upon computing thepositions of the external holes and the effect of the nonlinearterms we write a linear integral equation equivalent to ABA





(

119906119896+ 1198942

119906119896minus 1198942

)

119871

(

1 + 1198922

2119909minus

(119906119896)2

1 + 11989222119909

+(119906

119896)2)

119871

=

119904

prod

119895=1

119895 =119896

119906119896minus 119906

119895minus 119894

119906119896minus 119906

119895+ 119894

(

1 minus 1198922

2119909+

(119906119896) 119909

minus

(119906119895)

1 minus 11989222119909

minus(119906

119896) 119909

+(119906

119895)

)

2

sdot 1198902119894120579(119906119896 119906119895)

(7)

where

119909plusmn

(119906119896) = 119909 (119906

119896plusmn

119894

2

)

119909 (119906) =

119906

2

[

[

1 +radic1 minus

21198922

1199062

]

]

120582 = 81205872

1198922

(8)


120579 (119906 V) =infin

sum

119903=2

infin

sum

]=0120573119903119903+1+2] (119892)

sdot [119902119903(119906) 119902

119903+1+2] (V) minus 119902119903 (V) 119902119903+1+2] (119906)]

(9)

the functions 120573119903119903+1+2](119892) = 119892


119888119903119903+1+2](119892) being

120573119903119903+1+2] (119892)

= 2

infin

sum

120583=]

1198922119903+2]+2120583

2119903+120583+] (minus1)

119903+120583+1 (119903 minus 1) (119903 + 2])2120583 + 1

sdot (

2120583 + 1

120583 minus 119903 minus ] + 1)(

2120583 + 1

120583 minus ])120577 (2120583 + 1)

(10)

and 119902119903(119906)

119902119903(119906) =

119894

119903 minus 1

[(

1

119909+(119906)

)

119903minus1

minus (

1

119909minus(119906)

)

119903minus1

] (11)





119876119903(119892 119871 119904) =

119904

sum

119896=1

119902119903(119906

119896) (12)


120574 (119892 119871 119904) = 1198922

1198762(119892 119871 119904) (13)


119885 (119906) = Φ (119906) minus

119904

sum

119896=1

120601 (119906 119906119896) (14)

where

Φ (119906) = Φ0(119906) + Φ

119867(119906)

120601 (119906 V) = 1206010(119906 minus V) + 120601

119867(119906 V)

(15)

with

Φ0(119906) = minus2119871 arctan 2119906

Φ119867(119906) = minus119894119871 ln(

1 + 1198922

2119909minus

(119906)2

1 + 11989222119909

+(119906)

2)


120601119867(119906 V)

= minus2119894 [ln(1 minus 119892

2

2119909+

(119906) 119909minus

(V)1 minus 119892

22119909

minus(119906) 119909

+(V)

) + 119894120579 (119906 V)]

(16)



4

119906

+

21198941198922

119906

(

1

119909minus(V)

minus

1

119909+(V)

)

+ 119874(

1

1199063)

(17)




119885 (119906) 997888rarr ∓ (119871 + 119904) 120587 +

119871 + 2119904 + 120574 (119892 119871 119904)

119906

+ 119874(

1

1199063)

(18)








= minus119909119871 with 119909

119871gt 119887




119904

sum

119896=1

119874 (119906119896) +

119871

sum

ℎ=1

119874 (119909ℎ)

= minusint

+infin

minusinfin

119889V2120587

119874 (V) 1198851015840

(V)

+ int

+infin

minusinfin

119889V120587

119874 (V)119889


+)

]

(19)


1198851015840

(119906) = Φ1015840

(119906) + int

+infin

minusinfin

119889V2120587

119889

119889119906

120601 (119906 V) 1198851015840

(V)

+

119871

sum

ℎ=1

119889

119889119906

120601 (119906 119909ℎ)

minus int

+infin

minusinfin

119889V120587

119889

119889119906

120601 (119906 V)119889


+)

]

(20)


120590 (119906) = 1198851015840

(119906)

1198711015840

(119906) =

119889

119889119906

Im ln [1 + (minus1)119871 119890119894119885(119906minus1198940+)

]

(21)


+infin

minusinfin

119889119906119890minus119894119896119906


Φ0(119896) = minus

2120587119871119890minus|119896|2

119894119896

Φ119867(119896) =

2120587119871

119894119896

119890minus|119896|2

[1 minus 1198690(radic2119892119896)]

1206010(119896) =

2120587119890minus|119896|

119894119896

120601119867(119896 119905) = minus8119894120587

2119890minus(|119905|+|119896|)2

119896 |119905|

[

infin

sum

119903=1

119903 (minus1)119903+1

119869119903(radic2119892119896) 119869

119903(radic2119892119905)

sdot

1 minus sgn (119896119905)2

+ sgn (119905)infin

sum

119903=2

infin

sum

]=0119888119903119903+1+2] (119892) (minus1)

119903+]

sdot (119869119903minus1


(22)


(119896)

=

119894119896

1 minus 119890minus|119896|

Φ (119896) minus 2

119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(23)


119885 (119896)

=

1

1 minus 119890minus|119896|

Φ (119896) minus 2

119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

1

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) 119894119905 [

119885 (119905) minus 2 (119905)]

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(24)



119878 (119896) =

sinh (|119896| 2)120587 |119896|

(119896) +

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

120587119871

sinh (|119896| 2)(1 minus 119890

minus|119896|2

)

minus

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

[cos 119896119909ℎminus 1]

(25)


lim119896rarr0

119878 (119896) =

120574 (119892 119871 119904)

2

(26)


119878 (119896) =

119871

|119896|

(1 minus 1198690(radic2119892119896)) +

119894119896

1 minus 119890minus|119896|

sdot int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

sdot [

sum119871

ℎ=1(cos 119905119909

ℎminus 1) + 119871 (1 minus 119890

minus|119905|2

) minus (119894119905120587) (119905)

1 minus 119890minus|119905|

]

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

|119905|

2 sinh (|119905| 2)119878 (119905)

(27)


(119905 1199051015840

) =

2

1199051199051015840[

infin

sum

119899=1

119899119869119899(119905) 119869

119899(1199051015840

)

+ 2

infin

sum

119896=1

infin

sum

119897=0

(minus1)119896+119897

1198882119896+12119897+2

(119892) 1198692119896(119905) 119869

2119897+1(1199051015840

)]

(28)


int

+infin

minusinfin

119889119905120601119867(119896 119905) 119891 (119905)

= 81198941205872

1198922

int

+infin

0

119889119905119890minus(119905+119896)2

(radic2119892119896radic2119892119905) 119891 (119905)

119891 (119905) = 119891 (minus119905)

(29)


119878 (119896) =

119871

119896

(1 minus 1198690(radic2119892119896))

minus 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

sdot [

120587119905

sinh (1199052)119878 (119905) minus

2119894119905

1 minus 119890minus119905 (119905) +

119894119905

1 minus 119890minus119905Φ0(119905)

+ (

119894119905


1206010(119905) + 2120587)

119871

sum

ℎ=1

119890119894119905119909ℎ]

(30)




119885 (119909ℎ) = 120587 (2ℎ + 1 minus 119871) ℎ = 1 119871 minus 2 (31)


119885 (119909119871minus1) = minus119885 (119909

119871) = 120587 (119904 + 119871 minus 1) (32)





119897lt 119906

1198971015840 if 119897 lt 1198971015840


119904119906

1(119885(119906

119904119906

1) = ∓120587(119904+


119871minus1 we


119885 (plusmn119887) = ∓120587 (119904 + 119871 minus 2) (33)




119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)




sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)


119906119897 where 119906

119897119904 = 119906



119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)


minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)


1= minus119906

119904to the last 119906


term 1(119906119897minus119906

119904)+1(119906

119897+119906


1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)


119897


119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)


0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)


0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)



120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







(

119906119896+ 1198942

119906119896minus 1198942

)

119871

(

1 + 1198922

2119909minus

(119906119896)2

1 + 11989222119909

+(119906

119896)2)

119871

=

119904

prod

119895=1

119895 =119896

119906119896minus 119906

119895minus 119894

119906119896minus 119906

119895+ 119894

(

1 minus 1198922

2119909+

(119906119896) 119909

minus

(119906119895)

1 minus 11989222119909

minus(119906

119896) 119909

+(119906

119895)

)

2

sdot 1198902119894120579(119906119896 119906119895)

(7)

where

119909plusmn

(119906119896) = 119909 (119906

119896plusmn

119894

2

)

119909 (119906) =

119906

2

[

[

1 +radic1 minus

21198922

1199062

]

]

120582 = 81205872

1198922

(8)


120579 (119906 V) =infin

sum

119903=2

infin

sum

]=0120573119903119903+1+2] (119892)

sdot [119902119903(119906) 119902

119903+1+2] (V) minus 119902119903 (V) 119902119903+1+2] (119906)]

(9)

the functions 120573119903119903+1+2](119892) = 119892


119888119903119903+1+2](119892) being

120573119903119903+1+2] (119892)

= 2

infin

sum

120583=]

1198922119903+2]+2120583

2119903+120583+] (minus1)

119903+120583+1 (119903 minus 1) (119903 + 2])2120583 + 1

sdot (

2120583 + 1

120583 minus 119903 minus ] + 1)(

2120583 + 1

120583 minus ])120577 (2120583 + 1)

(10)

and 119902119903(119906)

119902119903(119906) =

119894

119903 minus 1

[(

1

119909+(119906)

)

119903minus1

minus (

1

119909minus(119906)

)

119903minus1

] (11)





119876119903(119892 119871 119904) =

119904

sum

119896=1

119902119903(119906

119896) (12)


120574 (119892 119871 119904) = 1198922

1198762(119892 119871 119904) (13)


119885 (119906) = Φ (119906) minus

119904

sum

119896=1

120601 (119906 119906119896) (14)

where

Φ (119906) = Φ0(119906) + Φ

119867(119906)

120601 (119906 V) = 1206010(119906 minus V) + 120601

119867(119906 V)

(15)

with

Φ0(119906) = minus2119871 arctan 2119906

Φ119867(119906) = minus119894119871 ln(

1 + 1198922

2119909minus

(119906)2

1 + 11989222119909

+(119906)

2)


120601119867(119906 V)

= minus2119894 [ln(1 minus 119892

2

2119909+

(119906) 119909minus

(V)1 minus 119892

22119909

minus(119906) 119909

+(V)

) + 119894120579 (119906 V)]

(16)



4

119906

+

21198941198922

119906

(

1

119909minus(V)

minus

1

119909+(V)

)

+ 119874(

1

1199063)

(17)




119885 (119906) 997888rarr ∓ (119871 + 119904) 120587 +

119871 + 2119904 + 120574 (119892 119871 119904)

119906

+ 119874(

1

1199063)

(18)








= minus119909119871 with 119909

119871gt 119887




119904

sum

119896=1

119874 (119906119896) +

119871

sum

ℎ=1

119874 (119909ℎ)

= minusint

+infin

minusinfin

119889V2120587

119874 (V) 1198851015840

(V)

+ int

+infin

minusinfin

119889V120587

119874 (V)119889


+)

]

(19)


1198851015840

(119906) = Φ1015840

(119906) + int

+infin

minusinfin

119889V2120587

119889

119889119906

120601 (119906 V) 1198851015840

(V)

+

119871

sum

ℎ=1

119889

119889119906

120601 (119906 119909ℎ)

minus int

+infin

minusinfin

119889V120587

119889

119889119906

120601 (119906 V)119889


+)

]

(20)


120590 (119906) = 1198851015840

(119906)

1198711015840

(119906) =

119889

119889119906

Im ln [1 + (minus1)119871 119890119894119885(119906minus1198940+)

]

(21)


+infin

minusinfin

119889119906119890minus119894119896119906


Φ0(119896) = minus

2120587119871119890minus|119896|2

119894119896

Φ119867(119896) =

2120587119871

119894119896

119890minus|119896|2

[1 minus 1198690(radic2119892119896)]

1206010(119896) =

2120587119890minus|119896|

119894119896

120601119867(119896 119905) = minus8119894120587

2119890minus(|119905|+|119896|)2

119896 |119905|

[

infin

sum

119903=1

119903 (minus1)119903+1

119869119903(radic2119892119896) 119869

119903(radic2119892119905)

sdot

1 minus sgn (119896119905)2

+ sgn (119905)infin

sum

119903=2

infin

sum

]=0119888119903119903+1+2] (119892) (minus1)

119903+]

sdot (119869119903minus1


(22)


(119896)

=

119894119896

1 minus 119890minus|119896|

Φ (119896) minus 2

119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(23)


119885 (119896)

=

1

1 minus 119890minus|119896|

Φ (119896) minus 2

119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

1

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) 119894119905 [

119885 (119905) minus 2 (119905)]

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(24)



119878 (119896) =

sinh (|119896| 2)120587 |119896|

(119896) +

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

120587119871

sinh (|119896| 2)(1 minus 119890

minus|119896|2

)

minus

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

[cos 119896119909ℎminus 1]

(25)


lim119896rarr0

119878 (119896) =

120574 (119892 119871 119904)

2

(26)


119878 (119896) =

119871

|119896|

(1 minus 1198690(radic2119892119896)) +

119894119896

1 minus 119890minus|119896|

sdot int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

sdot [

sum119871

ℎ=1(cos 119905119909

ℎminus 1) + 119871 (1 minus 119890

minus|119905|2

) minus (119894119905120587) (119905)

1 minus 119890minus|119905|

]

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

|119905|

2 sinh (|119905| 2)119878 (119905)

(27)


(119905 1199051015840

) =

2

1199051199051015840[

infin

sum

119899=1

119899119869119899(119905) 119869

119899(1199051015840

)

+ 2

infin

sum

119896=1

infin

sum

119897=0

(minus1)119896+119897

1198882119896+12119897+2

(119892) 1198692119896(119905) 119869

2119897+1(1199051015840

)]

(28)


int

+infin

minusinfin

119889119905120601119867(119896 119905) 119891 (119905)

= 81198941205872

1198922

int

+infin

0

119889119905119890minus(119905+119896)2

(radic2119892119896radic2119892119905) 119891 (119905)

119891 (119905) = 119891 (minus119905)

(29)


119878 (119896) =

119871

119896

(1 minus 1198690(radic2119892119896))

minus 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

sdot [

120587119905

sinh (1199052)119878 (119905) minus

2119894119905

1 minus 119890minus119905 (119905) +

119894119905

1 minus 119890minus119905Φ0(119905)

+ (

119894119905


1206010(119905) + 2120587)

119871

sum

ℎ=1

119890119894119905119909ℎ]

(30)




119885 (119909ℎ) = 120587 (2ℎ + 1 minus 119871) ℎ = 1 119871 minus 2 (31)


119885 (119909119871minus1) = minus119885 (119909

119871) = 120587 (119904 + 119871 minus 1) (32)





119897lt 119906

1198971015840 if 119897 lt 1198971015840


119904119906

1(119885(119906

119904119906

1) = ∓120587(119904+


119871minus1 we


119885 (plusmn119887) = ∓120587 (119904 + 119871 minus 2) (33)




119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)




sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)


119906119897 where 119906

119897119904 = 119906



119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)


minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)


1= minus119906

119904to the last 119906


term 1(119906119897minus119906

119904)+1(119906

119897+119906


1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)


119897


119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)


0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)


0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)



120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119885 (119906) 997888rarr ∓ (119871 + 119904) 120587 +

119871 + 2119904 + 120574 (119892 119871 119904)

119906

+ 119874(

1

1199063)

(18)








= minus119909119871 with 119909

119871gt 119887




119904

sum

119896=1

119874 (119906119896) +

119871

sum

ℎ=1

119874 (119909ℎ)

= minusint

+infin

minusinfin

119889V2120587

119874 (V) 1198851015840

(V)

+ int

+infin

minusinfin

119889V120587

119874 (V)119889


+)

]

(19)


1198851015840

(119906) = Φ1015840

(119906) + int

+infin

minusinfin

119889V2120587

119889

119889119906

120601 (119906 V) 1198851015840

(V)

+

119871

sum

ℎ=1

119889

119889119906

120601 (119906 119909ℎ)

minus int

+infin

minusinfin

119889V120587

119889

119889119906

120601 (119906 V)119889


+)

]

(20)


120590 (119906) = 1198851015840

(119906)

1198711015840

(119906) =

119889

119889119906

Im ln [1 + (minus1)119871 119890119894119885(119906minus1198940+)

]

(21)


+infin

minusinfin

119889119906119890minus119894119896119906


Φ0(119896) = minus

2120587119871119890minus|119896|2

119894119896

Φ119867(119896) =

2120587119871

119894119896

119890minus|119896|2

[1 minus 1198690(radic2119892119896)]

1206010(119896) =

2120587119890minus|119896|

119894119896

120601119867(119896 119905) = minus8119894120587

2119890minus(|119905|+|119896|)2

119896 |119905|

[

infin

sum

119903=1

119903 (minus1)119903+1

119869119903(radic2119892119896) 119869

119903(radic2119892119905)

sdot

1 minus sgn (119896119905)2

+ sgn (119905)infin

sum

119903=2

infin

sum

]=0119888119903119903+1+2] (119892) (minus1)

119903+]

sdot (119869119903minus1


(22)


(119896)

=

119894119896

1 minus 119890minus|119896|

Φ (119896) minus 2

119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

119894119896

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(23)


119885 (119896)

=

1

1 minus 119890minus|119896|

Φ (119896) minus 2

119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

1

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) 119894119905 [

119885 (119905) minus 2 (119905)]

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

119890119894119896119909ℎ

1206010(119896)

+

1

1 minus 119890minus|119896|

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

(24)



119878 (119896) =

sinh (|119896| 2)120587 |119896|

(119896) +

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

120587119871

sinh (|119896| 2)(1 minus 119890

minus|119896|2

)

minus

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

[cos 119896119909ℎminus 1]

(25)


lim119896rarr0

119878 (119896) =

120574 (119892 119871 119904)

2

(26)


119878 (119896) =

119871

|119896|

(1 minus 1198690(radic2119892119896)) +

119894119896

1 minus 119890minus|119896|

sdot int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

sdot [

sum119871

ℎ=1(cos 119905119909

ℎminus 1) + 119871 (1 minus 119890

minus|119905|2

) minus (119894119905120587) (119905)

1 minus 119890minus|119905|

]

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

|119905|

2 sinh (|119905| 2)119878 (119905)

(27)


(119905 1199051015840

) =

2

1199051199051015840[

infin

sum

119899=1

119899119869119899(119905) 119869

119899(1199051015840

)

+ 2

infin

sum

119896=1

infin

sum

119897=0

(minus1)119896+119897

1198882119896+12119897+2

(119892) 1198692119896(119905) 119869

2119897+1(1199051015840

)]

(28)


int

+infin

minusinfin

119889119905120601119867(119896 119905) 119891 (119905)

= 81198941205872

1198922

int

+infin

0

119889119905119890minus(119905+119896)2

(radic2119892119896radic2119892119905) 119891 (119905)

119891 (119905) = 119891 (minus119905)

(29)


119878 (119896) =

119871

119896

(1 minus 1198690(radic2119892119896))

minus 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

sdot [

120587119905

sinh (1199052)119878 (119905) minus

2119894119905

1 minus 119890minus119905 (119905) +

119894119905

1 minus 119890minus119905Φ0(119905)

+ (

119894119905


1206010(119905) + 2120587)

119871

sum

ℎ=1

119890119894119905119909ℎ]

(30)




119885 (119909ℎ) = 120587 (2ℎ + 1 minus 119871) ℎ = 1 119871 minus 2 (31)


119885 (119909119871minus1) = minus119885 (119909

119871) = 120587 (119904 + 119871 minus 1) (32)





119897lt 119906

1198971015840 if 119897 lt 1198971015840


119904119906

1(119885(119906

119904119906

1) = ∓120587(119904+


119871minus1 we


119885 (plusmn119887) = ∓120587 (119904 + 119871 minus 2) (33)




119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)




sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)


119906119897 where 119906

119897119904 = 119906



119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)


minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)


1= minus119906

119904to the last 119906


term 1(119906119897minus119906

119904)+1(119906

119897+119906


1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)


119897


119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)


0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)


0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)



120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878 (119896) =

sinh (|119896| 2)120587 |119896|

(119896) +

2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+

120587119871

sinh (|119896| 2)(1 minus 119890

minus|119896|2

)

minus

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1

[cos 119896119909ℎminus 1]

(25)


lim119896rarr0

119878 (119896) =

120574 (119892 119871 119904)

2

(26)


119878 (119896) =

119871

|119896|

(1 minus 1198690(radic2119892119896)) +

119894119896

1 minus 119890minus|119896|

sdot int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

sdot [

sum119871

ℎ=1(cos 119905119909

ℎminus 1) + 119871 (1 minus 119890

minus|119905|2

) minus (119894119905120587) (119905)

1 minus 119890minus|119905|

]

+

119894119896

1 minus 119890minus|119896|

int

+infin

minusinfin

119889119905

2120587

120601119867(119896 119905)

|119905|

2 sinh (|119905| 2)119878 (119905)

(27)


(119905 1199051015840

) =

2

1199051199051015840[

infin

sum

119899=1

119899119869119899(119905) 119869

119899(1199051015840

)

+ 2

infin

sum

119896=1

infin

sum

119897=0

(minus1)119896+119897

1198882119896+12119897+2

(119892) 1198692119896(119905) 119869

2119897+1(1199051015840

)]

(28)


int

+infin

minusinfin

119889119905120601119867(119896 119905) 119891 (119905)

= 81198941205872

1198922

int

+infin

0

119889119905119890minus(119905+119896)2

(radic2119892119896radic2119892119905) 119891 (119905)

119891 (119905) = 119891 (minus119905)

(29)


119878 (119896) =

119871

119896

(1 minus 1198690(radic2119892119896))

minus 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

sdot [

120587119905

sinh (1199052)119878 (119905) minus

2119894119905

1 minus 119890minus119905 (119905) +

119894119905

1 minus 119890minus119905Φ0(119905)

+ (

119894119905


1206010(119905) + 2120587)

119871

sum

ℎ=1

119890119894119905119909ℎ]

(30)




119885 (119909ℎ) = 120587 (2ℎ + 1 minus 119871) ℎ = 1 119871 minus 2 (31)


119885 (119909119871minus1) = minus119885 (119909

119871) = 120587 (119904 + 119871 minus 1) (32)





119897lt 119906

1198971015840 if 119897 lt 1198971015840


119904119906

1(119885(119906

119904119906

1) = ∓120587(119904+


119871minus1 we


119885 (plusmn119887) = ∓120587 (119904 + 119871 minus 2) (33)




119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)




sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)


119906119897 where 119906

119897119904 = 119906



119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)


minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)


1= minus119906

119904to the last 119906


term 1(119906119897minus119906

119904)+1(119906

119897+119906


1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)


119897


119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)


0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)


0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)



120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119909119871= minus119909

119871minus1

=

119904

radic2

[1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

+ 119874(

1

1199042)]

(34)




sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

(35)


119906119897 where 119906

119897119904 = 119906



119885 (119906119897119904)

= minus120587119871 sgn (119906119897) +

120574 (119892 119871 119904) + 119871

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

minus 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

+ 2 (119871 minus 2) [

120587

2

sgn (119906119897) minus

1

119906119897119904

] + 119874(

1

1199042)

(36)

where 119909ℎ= 119909

ℎ119904

120588 (119906) = minus

1

2120587119904

119889

119889119906

119885 (119906119904) (37)


minus 2

119904

sum

119896=1

arctan (119906119897minus 119906

119896) minus 2

119871minus2

sum

ℎ=1

arctan (119906119897minus 119909

ℎ)

+ 120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) + 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ)

=

1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(38)


1= minus119906

119904to the last 119906


term 1(119906119897minus119906

119904)+1(119906

119897+119906


1

119894

119904

sum

119896=1

119896 =119897

ln119906119897minus 119906

119896+ 119894

119906119897minus 119906

119896minus 119894

+

1

119894

119871minus2

sum

ℎ=1

ln119906119897minus 119909

ℎ+ 119894

119906119897minus 119909

ℎminus 119894

= 2int

119906119904

minus119906119904

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+

1

119904 (119906119897minus 119906

119904)

+

1

119904 (119906119897+ 119906

119904)

119874(

1

1199042)

(39)


119897


119885 (119906119897119904) = minus120587

119904

sum

119896=1

119896 =119897

sgn (119906119897minus 119906

119896) minus 120587

119871minus2

sum

ℎ=1

sgn (119906119897minus 119909

ℎ) (40)


0 = minus2120587 sgn (119906119897) +

4 minus 119871 + 120574 (119892 119871 119904)

119906119897119904

+ 2int

119887

minus119887

119889V120588 (V) 1198751

119906119897minus V

+

120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

+ 119874(

1

1199042)

(41)


0 = minus2120587 sgn (119906) + 2int119887

minus119887

119889V120588 (V) 1198751

119906 minus V(42)



120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

(43)


120587

119904

1205881015840

(119906119897) coth120587120588 (119906

119897)

=

1

119904

[

1

2119887 + 2119906119897

minus

1

2119887 minus 2119906119897

minus

2

119906119897

] + 119874(

1

1199042)

(44)


120588 (119906) =

1

120587

ln(119887 +radic119887

2

minus 1199062

119906

)

2

minus

(2 + 120574 (119892 119871 119904) minus 119871) 120575 (119906) + 120575 (119906 + 119887) + 120575 (119906 minus 119887)

2119904

+ 119874(

1

1199042)

(45)


int

119887

minus119887

119889119906120588 (119906) = minus

119885 (119887) minus 119885 (minus119887)

2120587119904

= 1 +

119871 minus 2

119904

(46)


119887 =

1

2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042) (47)


1= minus119906

119904to 119906

119904


119906119904= 119887 + 119874(

1

119904

) (48)


(119887)

1198851015840

(119887) sim

119885 (119887) minus 119885 (119906119904)

119887 minus 119906119904

sim

120587

119874 (1119904)

sim 119874 (119904) (49)




119871119904 119909

119871gt 119887



sum

119896=1

arctan (119906119904 minus 119906119896119904)

+

120574 (119892 119871 119904)

119906119904

+ 119874(

1

1199042)

= minus (119871 + 119904) 120587 sgn (119906) +119871 + 120574

119906119904

+

2

119904

119904

sum

119896=1

1

119906 minus 119906119896

+ 119874(

1

1199042)

(50)


2

119904

119904

sum

119896=1

1

119906 minus 119906119896

= 2int

119887

minus119887

119889V120588 (V)1


119871minus2

sum

ℎ=1

1

119906119904 minus 119909ℎ

+ 119874(

1

1199042)

= 2int

119887

minus119887

119889V120588 (V)1

119906 minus Vminus

2119871 minus 4

119906119904

+ 119874(

1

1199042)

(51)


int

119887

minus119887

119889V119906 minus V

ln(119887 +radic119887

2

minus V2

V)

2

= 119894120587 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

(52)


119885 (119906119904) = minus (119871 + 119904) 120587 sgn (119906) + 2

119906119904

minus

1

2119904

(

1

119906 + 119887

+

1

119906 minus 119887

)

+ 2119894 ln119894119906radic1 minus (119887

2

1199062

) + 119887

119894119906radic1 minus (119887

2

1199062

) minus 119887

+ 119874(

1

1199042)

(53)




119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119909119871= radic2 119887 + 119874(

1

1199042) 997904rArr

119909119871=

1

radic2

(1 +

119871 minus 1 + 120574 (119892 119871 119904)

2119904

) + 119874(

1

1199042)

(54)



1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

= 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042)

(55)



119873119871 (119896)

= 1198922

int

+infin

0

119889119905

120587

119890minus1199052

(radic2119892119896radic2119892119905)

2119894119905

1 minus 119890minus119905 (119905)

(56)


119868120572


+infin

0

119889119905

2120587

cos 1199051199062119894119905119890minus120572119905

1 minus 119890minus119905 (119905)

= int

+infin

minusinfin

119889V119894120587

[1205951015840


(57)


119873119871 (119906) = 2int

+infin

0

119889119896

2120587

cos 119896119906119873119871 (119896)

= minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) (58)

where

119870(

119906

radic2119892

Vradic2119892

) = 81198922

int

+infin

0

119889119896

2120587

int

+infin

0

119889119905

2120587

cos 119896119906


(59)


out(119906) where

119868120572

in (119906) = int119887

minus119887

119889V119894120587

[1205951015840


sdot 119871 (V)

119868120572

out (119906)

= int

|V|gt119887

119889V119894120587

[1205951015840


sdot 119871 (V)

(60)


119868120572

in (119906) = minus1198941198612 (1

2

)

1205951015840

(120572 minus 119894119906 + 119894119887) minus 1205951015840

(120572 + 119894119906 minus 119894119887) minus 1205951015840

(120572 minus 119894119906 minus 119894119887) + 1205951015840

(120572 + 119894119906 + 119894119887)

1198851015840(119887)

+ 119874(

1

1198851015840(119887)

3)

=

21198612(12)

1198851015840(119887)

[

119906 minus 119887

1205722+ (119906 minus 119887)

2minus

119906 + 119887

1205722+ (119906 + 119887)

2] + 119874(

1

1198851015840(119887)

3)

(61)



2

) andconsequently

119868120572

(119906)

= int

|V|gt119887

119889V119894120587

[1205951015840



+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+ 119874(

1

1199042)

(62)


119868120572

(119906) = minus

4

120587

int

+infin

119887

119889VV119871 (V) + 119874(

1

1199042) |119906| ≪ 119904 (63)


119868120572

(119906) =

4

120587

ln 119887119871 (119887) + 4

120587

int

+infin

119887

119889V ln V1198711015840 (V)

+ 119874(

1

1199042) |119906| ≪ 119904

(64)



+infin

119887

119889V2120587

ln V1198851015840

(V) + int+infin

119887

119889V120587

ln V1198711015840 (V) (65)

to obtain

119868120572

(119906) = 4 ln119909119871+

2

120587

int

+infin

119887

119889V ln V1198851015840

(V) (66)



1198851015840

(V) = minus4119887

V1

radicV2 minus 1198872+ 119874(

1

1198873) V gt 119887 (67)


minus

8119887

120587

int

+infin

119887

119889Vln VV

1



119874(1119904) we obtain

119868120572

(119906) = minus2 ln 2 + 119874( 11199042) |119906| ≪ 119904 (69)


119870(

119906

radic2119892

Vradic2119892

) = minus

1

1205872ln[1 minus

1198924

4119909 (119906)2

119909 (V)2]

|119906| |V| ge radic2119892

119870(

119906

radic2119892

Vradic2119892

) = minus

1

21205872

sdot ln([1 +1198922

1198902119894 arcsin(119906radic2119892)

2119909 (V)2]

sdot [1 +

1198922


2119909 (V)2])


(70)

Therefore

119873119871 (119906) = minusint

+infin

0

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

= 2 ln 2intΛ

0

119889V119870(119906

radic2119892

Vradic2119892

)

minus int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V)

+ 119874(

1

1199042)

(71)



119870(

119906

radic2119892

Vradic2119892

) sim

1

V2

11986812

(V) sim1

V

(72)



int

+infin

Λ

119889V119870(119906

radic2119892

Vradic2119892

) 11986812

(V) sim1

Λ2sim

1

1199042

(73)


119873119871 (119896) = 21198922 ln 2 (radic2119892119896 0) + 119874( 1

1199042) (74)





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878 (119896) = 41198922 ln 119904 (radic2119892119896 0)

+ 41198922

int

+infin

0

119889119905

119890119905minus 1

lowast

(radic2119892119896radic2119892119905)

+

21198922

119904

(119871 + 120574 (119892 119871 119904) minus 1) (radic2119892119896 0)

+

119871

119896

[1 minus 1198690(radic2119892119896)] + 4119892

2

120574119864 (radic2119892119896 0)

+ 1198922

(119871 minus 2)

sdot int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

1 minus 1198901199052

sinh (1199052)

minus 1198922

int

+infin

0

119889119905 (radic2119892119896radic2119892119905)

sum119871minus2

ℎ=1[cos 119905119909

ℎminus 1]

sinh (1199052)

minus 1198922

int

+infin

0

119889119905119890minus1199052

(radic2119892119896radic2119892119905)

119905

sinh (1199052)119878 (119905)

+ 119874 (119904minus1


(75)

where lowast

(119905 1199051015840

) = (119905 1199051015840


minus1


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)119899+

infin

sum

119899=minus1

119878(119899)

(119896)

119904 (ln 119904)119899

+ 119874 (119904minus1


(76)


119909ℎ=

infin

sum

119899=1

(120572119899ℎ+

119899ℎ

119904



120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(78)





119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

(79)


120574(minus1)

(119892 119871) =

1

2

[119891 (119892)]2

120574(0)

(119892 119871) =

1

2

119891 (119892) [119871 minus 1 + 119891sl (119892 119871)]

(80)


(119892 119871)


119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)] (ln 119904)minus119899 997904rArr

120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(81)


119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 L) = 2120587119878(1)

119901(119892)

119871minus2

sum

ℎ=1

1205721ℎ1ℎ+

119878(minus1)

119901(119892)

2

sdot 120574(2)

(119892 119871)

119878(3)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

119871minus2

sum

ℎ=1

(1205722ℎ1ℎ+ 120572

1ℎ2ℎ)

+

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)


119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119878(4)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ3ℎ+ 120572

2ℎ2ℎ+ 120572

3ℎ1ℎ) minus

120587

3

119878(2)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ)3

1ℎ+

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871)

119878(5)

119901(119892 119871) = 2120587119878

(1)

119901(119892)

sdot

119871minus2

sum

ℎ=1

(1205721ℎ4ℎ+ 120572

2ℎ3ℎ+ 120572

3ℎ2ℎ+ 120572

4ℎ1ℎ) minus

120587

3

sdot 119878(2)

119901(119892)

119871minus2

sum

ℎ=1

(3 (1205721ℎ)2

1205722ℎ1ℎ+ (120572

1ℎ)3

2ℎ)

+

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871)

(82)

where 119878(1)

119901(119892) and 119878

(2)





119889119903

119889119906119903120590 (119906 = 0)

=

infin

sum

119899=minus1

(120590(119899)

119903+

(119899)

119903

119904

+ 119874(

1

1199042)) (ln 119904)minus119899

(83)


and 119898ℎ


119903and (119899)

119903 In


119898ℎ



(119899)

119901(119892 119871) (relations (D4))


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

= 2

(minus1)

2

120590(minus1)

2

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(84)


119878(119899)



119878(1)

119901(119892 119871) = 0

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus 119891 (119892) 119878(2)

119901(119892 119871)

119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 119878(2)

119901(119892 119871)

minus

3

2

119878(3)

119901(119892 119871) 119891 (119892)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892) 120574

(4)

(119892 119871)

2

minus 2119878(4)

119901(119892 119871) 119891 (119892)

minus

3

2

119878(3)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) minus

5

2

119878(5)

119901(119892 119871) 119891 (119892)

minus 2119878(4)

119901(119892 119871) (119891sl (119892 119871) + 119871 minus 1)

minus 119878(2)

119901(119892 119871) 120574

(2)

(119892 119871)

(85)


120574(1)

(119892 119871) = 0

120574(2)

(119892 119871) = minus

119891 (119892)

2

120574(2)

(119892 119871)

120574(3)

(119892 119871) = minus119891 (119892) 120574(3)

(119892 119871)

minus (119891sl (119892 119871) + 119871 minus 1) 120574(2)

(119892 119871)

120574(4)

(119892 119871) = minus

3

2

119891 (119892) 120574(4)

(119892 119871)

minus

3

2

(119891sl (119892 119871) + 119871 minus 1) 120574(3)

(119892 119871)

120574(5)

(119892 119871) = minus2119891 (119892) 120574(5)

(119892 119871)

minus 2 (119891sl (119892 119871) + 119871 minus 1) 120574(4)

(119892 119871)

minus (120574(2)

(119892 119871))

2

(86)






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

infin

sum

119899=minus1

120574(119899)

(119892 119871)

119904 (ln 119904)119899+ 119874 (119904

minus1


(87)


119875 (119904) = 119891 (119892) ln119862 (119904) + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln119862 (119904))119899

+ 119874(

1

1198622)

(88)


119862 (119904)2

= (119904 +

119871

2

minus 1) (119904 +

119871

2

) 997904rArr

119862 (119904) = 119904 +

119871 minus 1

2

+ 119874(

1

119904

)

(89)


120574 (119892 119871 119904) = 119891 (119892) ln 119904 + 119891sl (119892 119871) +infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899

+

ln 1199042119904

[119891 (119892)]2

+

1

2119904

119891 (119892) (119871 minus 1 + 119891sl (119892 119871))

+

119891 (119892)

2119904

infin

sum

119899=1

120574(119899)

(119892 119871)

(ln 119904)119899minus

infin

sum

119899=1

119899

sdot

120574(119899)

(119892 119871)

2119904 (ln 119904)119899+1[119891 (119892) ln 119904 + 119891sl (119892 119871) + 119871 minus 1

+

infin

sum

119898=1

120574(119898)

(119892 119871)

(ln 119904)119898] + 119874 (119904

minus1


(90)




S119901(119892 119904 119871) = P

119901(119904 +

1

2

120574 (119892 119904 119871)) (91)


P119901(119904) =

infin

sum

119899=0

119886(119899)

119901(ln119862 (119904))

119862 (119904)2119899

(92)


119876119903(119892 119871 119904) = 119875

119903(119904 +

1

2

120574 (119892 119904 119871)) (93)






(119899)




6 Conclusions









Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










Appendices



119878 (119896) =

119871

|119896|

[1 minus 1198690(radic2119892119896)] +

119894119896

2120587 |119896|

sdot 119890|119896|2

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ 120601119867(119896 119905)

+

119894119896

2120587 |119896|

119890|119896|2

int

+infin

minusinfin

119889119905

41205872

120601119867(119896 119905) [ (119905) minus 2119894119905 (119905)]

(A1)


119878 (0) = minus1198922

int

+infin

minusinfin

119889119905

81205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

4120587

119890119894119905119909ℎ119890 (119905)

(A2)

where

119890 (119905) =

2radic2120587

119892119905

119890minus|119905|2

1198691(radic2119892119905) (A3)


119890 (119906) = 1199022(119906) = 119894 [

1

119909+(119906)

minus

1

119909minus(119906)

] (A4)


1198922

sum119904

119896=1119890(119906


120574 (119892 119871 119904) = minus1198922

119871

sum

ℎ=1

119890 (119909ℎ) minus 119892

2

int

+infin

minusinfin

119889V2120587

119890 (V) 1198851015840

(V)

+ 1198922

int

+infin

minusinfin

119889V120587

119890 (V)119889


+)

]

(A5)


120574 (119892 119871 119904) = minus1198922

int

+infin

minusinfin

119889119905

41205872119890 (119905) [ (119905) minus 2119894119905 (119905)]

minus 1198922

119871

sum

ℎ=1

int

+infin

minusinfin

119889119905

2120587

119890119894119905119909ℎ119890 (119905)

(A6)


(0) = int

+infin

minusinfin

1198891199061198851015840

(119906) = minus2120587 (119871 + 119904) (A7)


119871 (119906) 997888rarr

119871 + 2119904 + 120574 (119892 119871 119904)

2119906

+ 119874(

1

1199063) (A8)


lim119896rarr0

plusmn (119896) = ∓

119894120587

2

[119871 + 2119904 + 120574 (119892 119871 119904)] (A9)


lim119896rarr0

plusmn (119896)

= lim119896rarr0

plusmnint

+infin

minusinfin


+)

]

= plusmn

120587

2119894

[119871 + 2119904 + 120574 (119892 119871 119904)]

(A10)





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

+ 119874(

1

1198872)

(B1)



119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119885 (119906) = minus119871120587 +

120574 + 119871

119906

minus 2

119904

sum

119896=1

arctan (119906 minus 119906119896)

minus 2

119871minus2

sum

ℎ=1


2119871 minus 4

119906

+ 119874(

1

1198872)

(B2)


119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) [1198851015840

(V) minus 21198711015840 (V)]

+ 119874(

1

1198872)

(B3)


of [82]

119885 (119906) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587


sdot 1198851015840

(V) + 2infin

sum

119896=0

(2120587)2119896+1

(2119896 + 2)

1198612119896+2

(

1

2

)

sdot [

119889

1198891199092119896+1


(119909))]

119909=119885(119887)

119909=119885(minus119887)

+ 119874(

1

1198872) = minus2120587 +

120574 minus 119871 + 4

119906

+ int

119887

minus119887

119889V120587

arctan (119906 minus V) 1198851015840

(V)

+

120587

61198851015840(119887)

[

1

1 + (119906 minus 119887)2minus

1

1 + (119906 + 119887)2]

+ 119874(

1

1198872)

(B4)



1198851015840

(119906) = minus

120574 minus 119871 + 4

1199062

+ int

119887

minus119887

119889V120587

1

1 + (119906 minus V)21198851015840

(V)

+ 119874(

1

1198872)

(B5)


1198851015840

(119906) = minus2 ln 119887 +radic119887

2minus 119906

2


2

+ 120587 (120574 minus 119871 + 4) 120575 (119906)

+ 119874(

1

1198872)

(B6)


int

119887

minus119887

119889V119911 minus V

ln 119887 +radic119887

2minus V2


= 119894120587 sgn (Im 119911) lnradic119887

2minus 119911

2minus 119887


2+ 119887

119911 notin [minus119887 119887]

(B7)


minus lnradic119887

2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198872) = minus2 ln 119887 +


2


2

+ 119874(

1

1198872)

(B8)



1198851015840


2minus (119906 + 119894)

2

+ 119887

radic1198872minus (119906 + 119894)

2

minus 119887

minus lnradic119887

2minus (119906 minus 119894)

2

+ 119887


2

minus 119887

+ 119874(

1

1198873)

119906 gt 119887

(B9)


1198851015840

(119906) = minus

4119887

119906

1


2

+ 119874(

1

1198873) (B10)




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119878 (119896) =

infin

sum

119901=1

S119901(119892 119871 119904)

119869119901(radic2119892119896)

119896

S119901(119892 119871 119904)

=

infin

sum

119899=minus1

[119878(119899)

119901(119892 119871) +

1

119904

119878(119899)

119901(119892 119871) + 119874(

1

1199042)]


120574(119899)

(119892 119871) = radic2119892119878(119899)

1(119892 119871)

120574(119899)

(119892 119871) = radic2119892 119878(119899)

1(119892 119871)

(C1)



119878(119899)

2119901minus1(119892 119871)

= radic21198921205751199011120574(119899)

(119892 119871)


+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901minus1

(radic2119892119905)

sinh (1199052)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119899)

119898(119892 119871)

119878(119899)

2119901(119892 119871)

= minus2119901int

+infin

0

119889119905

119905

(119899)

(119892 119905) 1198692119901(radic2119892119905)

sinh (1199052)

minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119899)

119898(119892 119871)

(C2)


119875 (119904 119892 119905) =

119871minus2

sum

ℎ=1

[cos 119905119909ℎminus 1]

=

infin

sum

119899=1

(119875(119899)

(119892 119905) +

(119899)

(119892 119905)

119904

) (ln 119904)minus119899

+ 119874 (119904minus1


(C3)


(119899)

(119892 119905) =

119899

sum

119903=1

119905119903 cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C4)


119878(119899)

119901(119892 119871) = minus2120587

119899

sum

119903=1

119878(1199032)

119901(119892) cos 120587119903

2

sdot sum

1198951 119895119899minus119903+1

sum119871minus2

ℎ=1(prod

119899minus119903+1

119898=1(120572

119898ℎ)119895119898)sum

119899minus119903+1

1198981015840=1

1198951198981015840 (

1198981015840ℎ120572

1198981015840ℎ)

prod119899minus119903+1

119898=1119895119898

+

119878(minus1)

119901(119892)

2

120574(119899)

(119892 119871)

119899minus119903+1

sum

119898=1

119895119898= 119903

119899minus119903+1

sum

119898=1

119898119895119898= 119899

(C5)


(see [52])

119878(119903)

2119901minus1(119892) = I

(119903)

2119901minus1(119892)


infin

sum

119898=1

1198852119901minus1119898

(119892) 119878(119903)

119898(119892)

119878(119903)

2119901(119892) = I

(119903)

2119901(119892) minus 4119901

infin

sum

119898=1

1198852119901119898

(119892) (minus1)119898

119878(119903)

119898(119892)

(C6)

with

I(119903)

119901(119892) = 119901int

+infin

0

119889ℎ

2120587

ℎ2119903minus1

119869119901(radic2119892ℎ)

sinh (ℎ2) (C7)




and 119898ℎ


120572119898ℎ

997888rarr 120572119898ℎ

+

119898ℎ

119904

120590(119899)

119903997888rarr 120590

(119899)

119903+

(119899)

119903

119904

(D1)


We obtain

119901+1ℎ

= minus

119901

sum

119903=1

[

[

120590(minus1)

119903

120590(minus1)

0

119901minus119903+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(minus1)

119903

120590(minus1)

0

minus

120590(minus1)

119903(minus1)

0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903+1

119901minus119903+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898


minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus

119901minus1

sum

119897=0

119901minus119897

sum

119903=1

[

[

120590(119897)

119903minus1

120590(minus1)

0

119901minus119903minus119897+1

sum

1198981015840=1

1198951198981015840 (

1198981015840ℎ)

(1205721198981015840ℎ)

+

(119897)

119903minus1

120590(minus1)

0

minus

120590(119897)


0

(120590(minus1)

0)

2

]

]

sum

1198951 119895119901minus119903minus119897+1

119901minus119903minus119897+1

prod

119898=1

(120572119898ℎ)119895119898

119895119898

1ℎ=

minus120587(minus1)

0(2ℎ + 1 minus 119871)

(120590(minus1)

0)

2

(D2)



119898=1119895119898= 119903 + 1 sum119901minus119903+1

119898=1119898119895

119898= 119901 + 1 while the


119898=1119895119898=

119903 sum119901minus119903minus119897+1

119898=1119898119895

119898= 119901 minus 119897 The first

119898ℎare

1ℎ= minus

120587 (2ℎ + 1 minus 119871) (minus1)

0

(120590(minus1)

0)

2

2ℎ= minus

120587 (2ℎ + 1 minus 119871) (0)

0

(120590(minus1)

0)

2

+

2120587 (2ℎ + 1 minus 119871) 120590(0)

0(minus1)

0

(120590(minus1)

0)

3

3ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

2120590(0)

0(0)

0

(120590(minus1)

0)

3minus

3 (120590(0)

0)

2

(minus1)

0

(120590(minus1)

0)

4

]

]

minus

1205873

(2ℎ + 1 minus 119871)3

6

[

[

(minus1)

2

(120590(minus1)

0)

4minus

4120590(minus1)

0(minus1)

0

(120590(minus1)

0)

5

]

]

4ℎ= 120587 (2ℎ + 1 minus 119871)

[

[

4 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

5

minus

3 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

4+

2120590(2)

0(minus1)

0

(120590(minus1)

0)

3minus

(2)

0

(120590(minus1)

0)

2

]

]

+

21205873

(2ℎ + 1 minus 119871)3

3

[

[

minus

5120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

6

+

120590(0)

0(minus1)

2

(120590(minus1)

0)

5+

120590(minus1)

2(0)

0

(120590(minus1)

0)

5+

120590(0)

2(minus1)

0

(120590(minus1)

0)

5

minus

(0)

2

4 (120590(minus1)

0)

4

]

]

(D3)


119901(119892 119871) in terms of 120590(119899)

119903and (119899)

119903 for 119899 = 2 5 we obtain

119878(2)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(2)

(119892 119871) minus

21205873

3

119878(1)

119901(119892)

sdot

(minus1)

0

(120590(minus1)

0)


119878(3)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(3)

(119892 119871) +

21205873

3

sdot

119878(1)

119901(119892)

(120590(minus1)

0)

3[

3120590(0)

0(minus1)

0

120590(minus1)

0

minus (0)

0] (119871 minus 1) (119871 minus 2) (119871

minus 3)

119878(4)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(4)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

4

1205872

6120590(minus1)

0

[119878(1)

119901(119892)(

5120590(minus1)

2(minus1)

0

120590(minus1)

0

minus (minus1)

2) +

(minus1)

0119878(2)

119901(119892)]

(5 + 3119871 (119871 minus 4))

5

+ 3120590(0)

0119878(1)

119901(119892) [

(0)

0minus

2120590(0)

0(minus1)

0

120590(minus1)

0

] (119871 minus 1) (119871


119878(5)

119901(119892 119871) =

119878(minus1)

119901(119892)

2

120574(5)

(119892 119871) +

21205873

3

sdot

1

(120590(minus1)

0)

3

1205872

2 (120590(minus1)

0)

2

[

[

119878(1)

119901(119892)(

120590(0)

2(minus1)

0

120590(minus1)

0

+

120590(0)

0(minus1)

2

120590(minus1)

0

+

120590(minus1)

2(0)

0

120590(minus1)

0

minus

(minus1)

2

5

minus

6120590(minus1)

2120590(0)

0(minus1)

0

(120590(minus1)

0)

2) + 119878

(2)

119901(119892)(5

(0)

0

minus

120590(0)

0(minus1)

0

120590(minus1)

0

)]

]

(5 + 3119871 (119871 minus 4))

3

+ 119878(1)

119901(119892)

[

[

minus(2)

0

+

3120590(2)

0(minus1)

0

120590(minus1)

0

minus

6 (120590(0)

0)

2

(0)

0

(120590(minus1)

0)

2

+

10 (120590(0)

0)

3

(minus1)

0

(120590(minus1)

0)

3

]

]


(D4)


E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




E Useful Relations


and (119899)119903




(2)

(119892 119905) = minus2

(minus1)

0

120590(minus1)

0

119875(2)

(119892 119905) (E1)


119878 (119896) =

infin

sum

119899=minus1

119878(119899)

(119896) (ln 119904)minus119899 +infin

sum

119899=minus1

119878(119899)

(119896)

(ln 119904)minus119899

119904

+ 119874 (119904minus1


(E2)


119878(minus1)

(119896) =

119891 (119892)

2

119878(minus1)

(119896)

119878(0)

(119896) =

119891sl (119892 119871) + 119871 minus 1

2

119878(minus1)

(119896)

119878(2)

(119896) =

120574(2)

(119892 119871)

2

119878(minus1)

(119896) minus 119891 (119892) 119878(2)

(119896)

(E3)


(119896) = minus

2120587119871119890minus|119896|2

1 minus 119890minus|119896|

+

2120587119871119890minus|119896|

1 minus 119890minus|119896|

+

2120587119890minus|119896|

1 minus 119890minus|119896|

119871

sum

ℎ=1


2119894119896119890minus|119896|

1 minus 119890minus|119896|

(119896)

+ 119866 (119896)

(E4)

where

119866 (119896) =

120587 |119896|

sinh (|119896| 2)119878 (119896) (E5)


120590 (119906) = 119871 [120595(

1

2

minus 119894119906) + 120595(

1

2

+ 119894119906)]



119871minus 119894119906)


119871+ 119894119906)

minus [2 ln 2 + 119874(1199062

1199042)]

+ int

+infin

minusinfin

119889119896119890119894119896119906

119890minus|119896|

1 minus 119890minus|119896|

119875 (119904 119892 119896) + 119866 (119906)

(E6)



119904

minus 2

119891sl (119892 119871) + 119871 minus 1

119904

minus 2

infin

sum

119899=1

120574(119899)

(119892 119871)

119904

(ln 119904)minus119899

+

infin

sum

119899=1

int

+infin

minusinfin

119889119896

119890minus|119896|

1 minus 119890minus|119896|

(119875(119899)

(119892 119896) +

(119899)

(119892 119896)

119904

)


(E7)


119891 (119892) = 2

(minus1)

0

120590(minus1)

0

119891sl (119892 119871) = 2(0)

0

120590(minus1)

0


120574(2)

(119892 119871) = 2

(2)

0

120590(minus1)

0

+ 4

120590(2)

0(minus1)

0

(120590(minus1)

0)

2

(E8)


119891 (119892) = 2

(minus1)

2

120590(minus1)

2

(E9)


119899 = 1 2 3 4 5




119899 = 1 2 3 4 are shown in [87]

119878(1)

119901(119892 119871) = 0 (F1)

119878(2)

119901(119892 119871) =

1205873

3 (120590(minus1)

0)


(1)

119901(119892) (F2)


119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119878(3)

119901(119892 119871) = minus2

1205873

120590(0)

0

3 (120590(minus1)

0)


sdot 119878(1)

119901(119892)

(F3)

119878(4)


[

[

1205872

(120590(0)

0)

2

2 (120590(minus1)

0)

4

minus

1205874

120590(minus1)

2

90 (120590(minus1)

0)

5(5 + 3119871 (119871 minus 4))

]

]

119878(1)

119901(119892)

minus

1205874

360 (120590(minus1)

0)

4(5 + 3119871 (119871 minus 4)) 119878

(2)

119901(119892)

(F4)

119878(5)


sdot

[

[

(

51205875

3

120590(minus1)

2120590(0)

0

(120590(minus1)

0)

6minus

1205875

3

120590(0)

2

(120590(minus1)

0)

5)

sdot

(5 + 3119871 (119871 minus 4))

15

+ (minus

41205873

(120590(0)

0)

3

3 (120590(minus1)

0)

5minus

21205873

120590(2)

0

3 (120590(minus1)

0)

3)]

]

119878(1)

119901(119892)

+

1205875

120590(0)

0

45 (120590(minus1)

0)

5119878(2)

119901(119892) (5 + 3119871 (119871 minus 4))

(F5)



Acknowledgments


Endnotes



Σ (119896) = 119867(119896)1003816100381610038161003816ln 119904 +

2119890minus|119896|

1 minus 119890minus|119896|

119894119896119867(119896)

10038161003816100381610038161003816ln 119904

(lowast)




119867(0) = (0) minus |



119885119899119898

(119892) = int

+infin

0

119889119905

119905

119869119899(radic2119892119905) 119869

119898(radic2119892119905)

119890119905minus 1

(lowastlowast)

References









5times119878










5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








5times119878



4CFT







5CFT






























5times119878






















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


















































1198601198891198785times 119878










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Reciprocity and Self-Tuning Relations without...

Documents

Transcript of Research Article Reciprocity and Self-Tuning Relations without...