research article integral and series representations of...

Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 181724 17 pageshttpdxdoiorg1011552013181724

Research ArticleIntegral and Series Representations of Riemannrsquos Zeta Functionand Dirichletrsquos Eta Function and a Medley of Related Results

Michael S Milgram

Consulting Physicist Geometrics Unlimited Ltd PO Box 1484 Deep River ON Canada K0J 1P0

Correspondence should be addressed to Michael S Milgram mikegeometrics-unlimitedcom

Received 22 September 2012 Accepted 20 December 2012

Academic Editor Alfredo Peris

Copyright copy 2013 Michael S MilgramThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Contour integral representations of Riemannrsquos Zeta function and Dirichletrsquos Eta (alternating Zeta) function are presented andinvestigated These representations flow naturally from methods developed in the 1800s but somehow they do not appear in thestandard reference summaries textbooks or literature Using these representations as a basis alternate derivations of known seriesand integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approachand results are developed that appear to be new

Dedicated to the memory of Leslie M Saundersdied September 8 1972 age 29

(Physics Today obituary issue of December 1972 page 63)

1 Introduction

Riemannrsquos Zeta function 120577(119904) and its sibling Dirichletrsquos(alternating zeta) function 120578(119904) play an important role inphysics complex analysis and number theory and have beenstudied extensively for several centuries In the same veinthe general importance of a contour integral representationof any function has been known for almost two centuriesso it is surprising that contour integral representations forboth 120577(119904) and 120578(119904) exist that cannot be found in any of themodern handbooks (NIST [1 Section 255(iii)] Abramowitzand Stegun [2 Chapter 23]) textbooks (Apostol [3 Chapter12] Olver [4 Chapter 82] Titchmarsh [5 Chapter 4]Whittaker and Watson [6 Section (1313)]) summaries(Edwards [7] Ivic [8 Chapter 4] Patterson [9] Srivastavaand Choi [10 11]) compendia (Erdelyi et al [12 Section 112]and Chapter 17) tables (Gradshteyn and Ryzhik [13] 9512Prudnikov et al [14 Appendix II7]) and websites [1 15ndash19] that summarize what is known about these functionsOne can only conclude that such representations generallydiscussed in the literature of the late 1800s have been long-buried and their significance has been overlooked bymodernscholars

It is the purpose of this work to disinter these represen-tations and revisit and explore some of the consequencesIn particular it is possible to obtain series and integralrepresentations of 120577(119904) and 120578(119904) that unify and generalizewell-known results in various limits and combinations andreproduce results that are only now being discovered andinvestigated by alternate means Additionally it is possible toexplore the properties of 120577(119904) in the complex plane and on thecritical line and obtain results that appear to be new Many ofthe results obtained are disparate and difficult to categorizein a unified manner but share the common theme that theyare all somehow obtained from a study of the revived integralrepresentations That is the unifying theme of this work Tomaintain a semblance of brevity most of the derivations areonly sketched with citations sufficient enough to allow thereader to reproduce any result for herhimself

2 Contour Integral Representations of120577(119904) and 120578(119904)

Throughout I will use 119904 = 120590 + 119894120588 where 119904 isin C (complex)120590 120588 isin R (reals) and 119899 and119873 are positive integers

2 Journal of Mathematics

The Riemann Zeta function 120577(119904) and the Dirichlet alter-nating zeta function 120578(119904) are well known and defined by(convergent) series representations

120577 (119904) equivinfin

sum119896=1

119896minus119904

120590 gt 1 (1)

120578 (119904) equiv minusinfin

sum119896=1

(minus1)119896

119896minus119904

120590 gt 0 (2)

with

120578 (119904) = (1 minus 21minus119904

) 120577 (119904) (3)

It is easily found by an elementary application of the residuetheorem that the following reproduces (1)

120577 (119904) = 120587infin

sum119896=1

residue 119905minus119904119890119894120587119905

sin (120587119905)

100381610038161003816100381610038161003816100381610038161003816119905=119896 120590 gt 1 (4)

and the following reproduces (2)

120578 (119904) = 120587infin

sum119896=1

residue(119905minus119904cot (120587119905) 119890119894120587119905)10038161003816100381610038161003816119905=119896 120590 gt 0 (5)

Convert the result (4) into a contour integral enclosing thepositive integers on the 119905-axis in a clockwise direction andprovided that 120590 gt 1 so that contributions from infinityvanish the contour may then be opened such that it stretchesvertically in the complex 119905-plane giving

120577 (119904) =119894

2int119888+119894infin

119888minus119894infin

119905minus119904cot (120587119905) 119889119905 0 lt 119888 lt 1 119888 isin R (6)

= minus1

2intinfin

minusinfin

(119888 + 119894119905)minus119904cot (120587 (119888 + 119894119905)) 119889119905 120590 gt 1 (7)

a result that cannot be found in any of the modern referenceworks cited (in particular [20]) It should be noted thatthis procedure is hardly novelmdashit is often employed in aneducational context (eg in the special case 119904 = 2) todemonstrate the utility of complex integration to evaluatespecial sums ([10 Section 41]) ([21 Lemma 21]) and formsthe basis for an entire branch of physics [22]

A similar application to (5) with some trivial trigonomet-ric simplification gives

120578 (119904) =minus119894

2int119888+119894infin


119905minus119904

sin (120587119905)119889119905 (8)

=1

2intinfin

minusinfin

(119888 + 119894119905)minus119904

sin (120587 (119888 + 119894119905))119889119905 (9)

Equation (9) reduces with 119888 = 12 to a Jensen result[23] presented (without proof) in 1895 and to the best ofmy knowledge reproduced again but only in the specialcase 119888 = 12 only once in the modern reference literature[17] In particular although Lindelof rsquos 1905 work [24] arrivesat a result equivalent to that obtained here attributing thetechnique to Cauchy (1827) it is performed in a general

context so that the specific results (7) and (9) never appearparticularly in that section of the work devoted to 120577(119904) Inmodern notation for ameromorphic function119891(119911) Lindelofwrites ([24 Equation III(4)]) in general

infin

sum]=minusinfin

119891 (]) = minus1

2120587119894∮120587 cot (120587119911) 119891 (119911) 119889119911 (10)

where the contour of integration encloses all the singularitiesof the integrand but never specifically identifies 119891(119911) =119911minus119904 Much later after the integral has been split into twoby the invocation of identities for cot(120587119911) he does makethis identification ([24 Equation 4III (6)]) arriving (with(3)) at ((26)mdashsee below) a now-well-known result therebybypassing the contour integral representation (7) In a similarvein Olver ([4 page 290]) reproduces (10) employing itto evaluate remainder terms the Abel-Plana formula and aJensen result but again never writes (7) explicitly Perhaps theomission of an explicit statement of (7) and (9) from ChapterIV of Lindelof [24] is why these forms seem to have vanishedfrom the historical record although citations to Jensen andLindelof abound inmodern summaries (Recently Srivastavaand Choi [11 pages 169ndash172] have reproduced Lindelof rsquosanalysis also failing to arrive explicitly at (7) although theydo write a general form of (8) ([11 Section 23 Equation(38)]) it is studied only for the case 119888 = 12) (Note added inproof after a draft copy of this paper was posted I was madeaware that Ruijsenaars has previously obtained a generalizedform of (7) for the Hurwitz zeta function ([25 Equation(422)]) He also notes (private communication) that he couldnot find prior references to his result in the literature)

At this point it is worthwhile to quote several relatedintegral representations that define some of the relevantand important functions of complex analysis starting withRiemannrsquos original 1859 contour integral representation of thezeta function [26]

120577 (119904) =Γ (1 minus 119904)

2120587119894∮

119905119904minus1

119890minus119905 minus 1119889119905 (11)

where the contour of integration encloses the negative 119905-axislooping from 119905 = minusinfinminus 1198940 to 119905 = minusinfin+ 1198940 enclosing the point119905 = 0

The equivalence of (11) and (1) is well described in texts(eg [11]) and is obtained by reducing three components ofthe contour in (11) A different analysis is possible however(eg [7 Section 16])mdashopen and translate the contour in (11)such that it lies vertically to the right of the origin in thecomplex 119905-plane (and thereby vanishes) with 120590 lt 0 andevaluate the residues of each of the poles of the integrandlying at 119905 = plusmn 2119899120587119894 to find

120577 (119904) =minus119894

2120587Γ (1 minus 119904) (2120587)

119904

times (exp (1198941205871199042) minus exp (minus119894120587119904

2))infin

sum119896=1

119896119904minus1

(12)

Journal of Mathematics 3

This can be further reduced giving the well-known func-tional equation for 120577(119904)

120577 (1 minus 119904) =2 cos (1205871199042) Γ (119904)

(2120587)119904120577 (119904) (13)

which is valid for all 119904 by analytic continuation from 120590 lt0 This demonstrates that (11) and (6) are fundamentallydifferent representations of 120577(119904) since further deformationsof (11) do not lead to (7)

Ephemerally at one time it was noted in [17] that thefollowing complex integral representation

120577 (119904) =120587

2 (119904 minus 1)intinfin

minusinfin

(12 + 119894119905)1minus119904

cosh2 (120587119905)119889119905 120590 gt 0 (14)

can be obtained by the invocation of the Cauchy-Schlomilchtransformation [27] In fact the representation (14) follows bythe simple expedient of integrating (7) by parts in the specialcase 119888 = 12 (see (35)) as also noted in [17] Another result

120577 (119904) =2119904minus1

(1 minus 21minus119904) Γ (119904 + 1)intinfin

0

119905119904

cosh2 (119905)119889119905 120590 gt minus1 (15)

obtained in [27] on the basis of the Cauchy-Schlomilchtransformation does not follow from (14) as erroneouslyclaimed at one time but later corrected in [17]

Based on (11) a number of other representations of 120577(119904)are well known From the tables cited the following are worthnoting here

([12 Equation 112(6)])

120577 (119904) =1

2Γ (119904) (1 minus 2minus119904)intinfin

0

119905119904minus1

sinh (119905)119889119905 120590 gt 1 (16)

([12 Equation 112(4)])

120577 (119904) =1

Γ (119904)intinfin

0

119905119904minus1

exp (119905) minus 1119889119905 120590 gt 1 (17)

and ([13 Equation 3527(1)])

120577 (119904) =1

4

(2119886)(119904+1)

Γ (119904 + 1)intinfin

0

119905119904

sinh2 (119886119905)119889119905 120590 gt 1 (18)

Byway of comparison Laplacersquos representation ([13 Equation8315(2)])

1

Γ (119904)=

1

2120587119894intinfin

minusinfin

119890(119888+119894119905)

(119888 + 119894119905)119904119889119905 (19)

defines the inverse Gamma function

3 Simple Reductions

31 The Case 0 le 119888 le 1 The simplest reductions of (7)and (9) are obtained by uniting the two halves of their rangecorresponding to 119905 lt 0 and 119905 gt 0 With the use of

(minus(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

) 119894 = 2(1198882

+ 1199052

)(1199042)

times sin(119904 arctan( 119905119888))

(20)

(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

= 2(1198882

+ 1199052

)(1199042)

cos(119904 arctan( 119905119888))

(21)

one finds (with 0 lt 119888 lt 1 and 0 le 119888 le 1 resp)

120577 (119904) = minusintinfin

0

(cos(119904 arctan( 119905119888)) cos (120587119888) sin (120587119888)

minus sinh (120587119905) cosh (120587119905) sin(119904 arctan( 119905119888)))

times ((1198882

+ 1199052

)1199042

(cosh2 (120587119905) minus cos2 (120587119888)))minus1

119889119905

120590 gt 1

(22)

120578 (119904) = intinfin

0

(minus sin(119904 arctan( 119905119888)) cos (120587119888) sinh (120587119905)

+ cos(119904 arctan( 119905119888)) sin (120587119888) cosh (120587119905))

times ((1198882

+ 1199052

)1199042


119889119905

(23)

the latter valid for all 119904 In the case that 119888 = 12 (22) reducesto

120577 (119904) = 2119904

intinfin

0

sinh (120587119905) sin (119904 arctan (2119905))(1 + 41199052)

1199042 cosh (120587119905)119889119905 120590 gt 1 (24)

equivalent to the well-known result (attributed to Jensenby Lindelof and a special case of Hermitersquos result for theHurwitz zeta function ([6 Section 132)]) (which also appearsas an exercise in many textbooksmdashfor example [11 Equation23(41)])

120577 (119904) =2(119904minus1)

119904 minus 1minus 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890minus120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 (25)

Similarly (23) reduces to the known ([24 IVIII(6)]) result

120578 (119904) = 2119904minus1

intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042 cosh (1205871199052)119889119905 (26)

Comparison of (24) and (25) immediately yields the knownresult ([13 Equation 3975(2)])

120577 (119904) =2(119904minus1)

1 minus 119904+ 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 120590 gt 1

(27)


Notice that (25) and (26) are valid for all 119904 Simple substitutionof 119888 rarr 0 in (23) gives

120578 (119904) = minus sin(1199041205872)intinfin

0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (28)

which after taking (3) into account leads to

120577 (119904) =sin (1199041205872)(21minus119904 minus 1)

intinfin

0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (29)

displaying the existence of the trivial zeros of 120577(119904) at 119904 = minus2119873since the integral in (29) is clearly finite at each of these valuesof 119904 The result (29) is equivalent to the known result (16)after taking (13) into account Other possibilities abound Forexample set 119888 = 13 in (22) to obtain

120577 (119904) = 3119904minus1

intinfin

0

(2 sin (119904 arctan (119905)) sinh(21205871199053

)

minusradic3 cos (119904 arctan (119905)))

times ((1 + 1199052

)1199042

(2 cosh (21205871199053

) + 1))minus1

119889119905

120590 gt 1

(30)

Further a novel representation can be found by noting that(9) is invariant under the transformation of variables 119905 rarr 119905 +120588 together with the choice 119888 = 120590 along with the requirementthat 0 le 120590 le 1 This effectively allows one to choose 119888 = 119904yielding the following complex representation for 120578(119904) as wellas the the corresponding representation for 120577(119904) because of(3) which is valid in the critical strip 0 le 120590 le 1

120578 (119904) =minus119894

2int119904+119894infin


119905minus119904

sin (120587119905)119889119905 (31)

=1

2intinfin

minusinfin

(119904 + 119894119905)minus119904

sin (120587 (119904 + 119894119905))119889119905 (32)

Because (20) and (21) are valid for 119888 isin C it follows that(23) with 119888 = 119904 is also a valid representation for 120578(119904) aswell as the the corresponding representation for 120577(119904) withrecourse to (3) again only valid in the critical strip 0 le 120590 le1 Another interesting variation is the choice 119888 = 1120588 in(23) valid for 120588 gt 1 again applicable to the correspondingrepresentation for 120577(119904) because of (3) Such a representationmay find application in the analysis of 120577(120590 + 119894120588) in theasymptotic limit 120588 rarr infin Finally the choice 119888 = 120590 0 le120590 le 1 leads to an interesting variation that is discussed inSection 9

32 Simple Recursion via Partial Differentiation As a func-tion of 119888 120577(119904) and 120578(119904) are constant as are their integralrepresentations over a range corresponding to 0 lt 119873 minus119888 lt 1 however the integrals change discontinuously at eachnew value of 119888 that corresponds to a new value of 119873 Thusthe operator 120597120597119888 acting on (7) and (9) vanishes except at

nonnegative integers 119888 = 119873 when it becomes indefiniteTherefore it is of interest to investigate this operation in theneighbourhood of 119888 = 1198732 as well as the limit 119888 rarr 119873

Applied to (9) (and equivalent to integration by parts) therequirement that

120597120578 (119904)

120597119888= 0 119888 =119873 (33)

and identification of one of the resulting terms yields

120578 (119904 + 1) = minus120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904 cos (120587 (119888 + 119894119905))

sin2 (120587 (119888 + 119894119905))119889119905

0 le 119888 lt 1

(34)

a new result with improved convergence at infinity Similarlyoperating with 120597120597119888 on (7) yields

120577 (119904 + 1) =120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904

sin2 (120587 (119888 + 119894119905))119889119905 0 lt 119888 lt 1 (35)

(It should be noted that Srivastava et al [20] study anincomplete form of (35) in order to obtain representationsfor sums both involving 120577(119899) and 120577(119899) by itself withoutcommenting on the significance of the contour integral withinfinite bounds (35)) Using (20) and (21) to reduce theintegration range of (34) gives a fairly lengthy result forgeneral values of 119888 If 119888 = 12 (34) becomes

120578 (119904 + 1) =2119904minus1120587

119904

times intinfin

0

(1 + 1199052

)minus1199042

sinh(1205871199052) sin (119904 arctan (119905))

times coshminus2 (1205871199052) 119889119905

(36)

which by letting 119904 rarr 119904 minus 1 can be rewritten as

120578 (119904 + 1) =2 (119904 minus 1) 120578 (119904)

119904+2119904minus1120587

119904

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(37)

and with regard to (3)

120577 (119904 + 1) =2 (119904 minus 1) (1 minus 21minus119904) 120577 (119904)

119904 (1 minus 2minus119904)+

2119904minus1120587

119904 (1 minus 2minus119904)

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(38)


Similarly ((37) and (38) are the first two entries in a familyof recursive relationships that will be discussed elsewhere(in preparation)) In the case that 119888 = 12 (35) reduces tothe well-known result ([24 Equation 4 III(5)] (attributed toJensen [23]))

120577 (119904 + 1) =1205872119904minus1

119904intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042cosh2 (1205871199052)119889119905 (39)

If 119888 rarr 0 in (34) (for the case 119888 = 0 in (35) see thefollowing section) one finds a finite result (also equivalentto integrating (28) by parts)

120578 (119904) =120587 sin (1199041205872)

119904 minus 1intinfin

0

1199051minus119904 cosh (120587119905)sinh2 (120587119905)

119889119905 120590 lt 0 (40)

and when 119888 = 1 one finds the unusual forms

120578 (119904 + 1) = 0 1 +119894120587119903

2119904

times int120587

0

(1 plusmn 119903119894119890119894120579)minus119904

cosh (120587119903119890119894120579) 119890119894120579

sinh2 (120587119903119890119894120579)119889120579

minus119903120587

119904intinfin

1

(1 + 1199032

1199052

)minus1199042

cos (119904 arctan (119903119905))

times cosh (120587119903119905) sinhminus2 (120587119903119905) 119889119905

(41)

which is valid for any 0 lt 119903 lt 1 and all 119904 (particularly 119903 = 1120587or 119903 = 12) after deformation of the contour of integrationin (34) either to the left or to the right of the singularity at 119905 =1 respectively belonging to a choice of plusmn where the symbol0 1means zero or one as the case may be

4 Special Cases

41 The Case 119888 = 0 With respect to (35) setting 119888 = 0together with some simplification gives

120577 (119904) = minus120587119904minus1

sin (1205871199042)119904 minus 1

intinfin

0

1199051minus119904

sinh2 (119905)119889119905 120590 lt 0 (42)

reproducing (18) with 119886 = 1 after taking (13) into accountThis same case (119888 = 0) naturally suggests that the integral in(6) be broken into two parts

120577 (119904) = 1205770(119904) + 120577

1(119904) (43)

where

1205770(119904) =

119894

2∮119894

minus119894

119905minus119904cot (120587119905) 119889119905 (44)

1205771(119904) =

119894

2int119894

minus119894infin

119905minus119904cot (120587119905) 119889119905 + 119894

2int119894infin

119894

119905minus119904cot (120587119905) 119889119905 (45)

such that the contour in (44) bypasses the origin to the rightIt is easy to show that the sum of the two integrals in (45)reduces to

1205771(119904) = sin(120587119904

2)intinfin

1

119905minus119904 coth (120587119905) 119889119905 120590 gt 1 (46)

=sin (1205871199042)119904 minus 1

+ sin(1205871199042)intinfin

1

119905minus119904119890(minus120587119905)

sinh (120587119905)119889119905 (47)

the latter being valid for all 119904 One of several commonlyused possibilities of reducing (44) is to apply ([28 Equation(5432)])

119905 cot (120587119905) = minus2

120587

infin

sum119896=0

1199052119896

120577 (2119896) |119905| lt 1 (48)

giving

1205770(119904) = minus

2

120587sin(120587119904

2)infin

sum119896=0

(minus1)119896120577 (2119896)

2119896 minus 119904 (49)

being convergent for all 119904 Such sums appear frequently in theliterature (eg [10 29] Section (32)) and have been studiedextensively when 119904 = minus119899 [10 30] In the case 119904 = 2119899 (49)reduces to 120577

0(119904) = 120577(119904) and (47) vanishes

Along with (42) the integral in (35) can be similarly splitusing the convergent series representation

1

sin2 (120587V)=

2

1205872

infin

sum119896=0

(2119896 minus 1) 120577 (2119896) V2119896minus2

|V| lt 1 (50)

where the Bernoulli numbers 1198612119896

appearing in the original([28 Equation (50610)]) are related to 120577(2119896) by

120577 (2119896) = minus1

2

(minus1)119896(2120587)21198961198612119896

Γ (2119896 + 1) (51)

Thus the representation

120577 (119904) = minus1

2

119894120587

(119904 minus 1)int119894

minus119894

1199051minus119904

sin2 (120587119905)119889119905

minus120587 sin (1205871199042)


1

1199051minus119904

sinh2 (120587119905)119889119905

(52)

becomes (formally)

120577 (119904) = minussin (1205871199042)119904 minus 1

times (2

120587

infin

sum119896=0

(minus1)119896 (2119896 minus 1) 120577 (2119896)

2119896 minus 119904

+120587intinfin

1

1199051minus119904

sinh2 (120587119905)119889119905)

(53)

after straightforward integration of (50) with 120590 lt 0 In(53) the integral converges and the formal sum conditionallyconverges or diverges according to several tests Howver


further analysis is still possible Following the method of[10 Section 33] applied to (49) together with [10 Equation34(15)] we have the following general result for |V| lt 1infin

sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)

so that (formally) after premultiplying (55) with 1V thenoperating with V2(120597120597V)

infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)

Notice that the right-hand side of (55) provides an analyticcontinuation of the left-hand side for |V| gt 1 and the right-hand side of (57) gives a regularization of the left-handside Putting together (53) and (57) incorporating the firstterm of (57) into the second term of (53) setting V = 119894and constraining 120590 gt 0 eventually yields a new integralrepresentation (compare with (121) [31 Equation (28)] and[25 Equation (480)])

120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)

which is valid for 0 lt 120590 lt 2 a nonoverlapping analyticcontinuation of (42) that spans the critical strip Additionallyusing (13) followed by replacing 119904 rarr 1 minus 119904 gives a secondrepresentation

120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)

valid for minus1 lt 120590 lt 1 a non-overlapping analyticcontinuation of (18) Over the critical strip 0 lt 120590 lt 1representations (58) and (59) are simultaneously true andaugment the rather short list of other representations thatshare this property ([32] as cited in [12] (unnumberedequations immediately following 112(1) also reproducedin [11 Equations 23(44)ndash(46)])) See Section 9 for furtherapplication of this result

42 The Case 119888 ge 1 Interesting results can be obtained bytranslating the contour in (6) and (9) and adding the residuesso-omitted From (1) and (2) define the partial sums

120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)

and adjust all the above results accordingly by redefining119888 rarr 119888

119873 (If 119888119873

= 119873 an extra half-residue is removed andthe contour deformed to pass to the right of the point 119905 = 119873)This gives for example the generalization of (6)

120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)

so (22) and (23) with 119888 rarr 119888119873become integral representa-

tions of the remainder of the partial sums 120577(119904) minus 120577119873(119904) and

120578(119904)minus120578119873(119904) the former corresponding to a special case of the

Hurwitz zeta function 120577(119904119873)In the case that 119888

119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1

2 (119904 minus 1)+ intinfin

0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)

with the latter two results being valid for all 119904 Neither (64) nor(66) appear in references (65) corresponds to a limiting caseof Hermitersquos representation of the Hurwitz zeta function ([12Equation 110(7)]) In analogy to the developments leading to(27) equations (64) (66) and (3) together with the followingmodified form of [13 Equation (9515)]

120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)

can be used to obtain the possibly new result

120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873

= 12 + 119873 corresponding to general partial sumstwo possibilities are apparent Based on (20) (21) (22) and

(23) (for comparison see also [4 Section 83] and [7 Section(64)] and Chapter 7) we find

120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904

119904 minus 1minus 2(

1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)

which together extend (24) and (26) respectively Applicationof (34) and (35) with (20) and (21) also with 119888

119873= 12 + 119873

gives the remainders

120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)

which are valid for all 119904 For the case 119888119873

= 119873 applying (62)and (63) gives the remainders

120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)

which extend (64) and (66) the latter being valid for all 119904It should be noted that (74) valid for 119904 isin C generalizeswell-established results (eg [13 Equation 3411(12)]) thatare only valid in the limiting case 119904 = 119899 (Although [13Equation 3411(21)] gives (with a missing minus sign) anintegral representation for finite sums such as those appearingin (74) specified to be valid only for integer exponents (119896119899)that particular result in [13] also appears to apply after thegeneralization 119899 rarr 119904) Omitting the integral term from(70) gives an upper bound for the remainder similarlyomitting the integral term in (74) gives a lower bound for the

remainder provided that 119904 isin R in both cases In additionmany of the above results in particular (70) and (74) appearto be new and should be compared to classical results such as[8 Theorem 18 page 23 and pages 99-100] and asymptoticapproximations such as [33 Equation (821)]

5 Series Representations

51 By Parts By simple manipulation of the integralrepresentations given it is possible to obtain new series


representations Starting from the integral representation(29) integrate by parts giving

120577 (119904) =sin (1205871199042) 1205872

(2(1minus119904) minus 1) (minus1199042 + 1)

times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)

times cosh (120587119905) sinhminus3 (120587119905) 119889119905 120590 lt 0

(76)

where the hypergeometric function arises because of [34Equation 6211(6)] Write the hypergeometric function as a(uniformly convergent) series thereby permitting the sumand integral to be interchanged apply a second integrationby parts and find

120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)

Identify the integral in (77) using (18) giving

120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 1

timesinfin

sum119896=0

2minus2119896120577 (minus119904 + 1 + 2119896) Γ (minus119904 + 2 + 2119896)

Γ (2119896 + 2)

(78)

which is valid for all values of 120590 by the ratio test and theprinciple of analytic continuation This result which appearsto be new can be rewritten by extracting the 119896 = 0 term andapplying (13) to yield

120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1

2minus2119896 120577 (minus119904 + 1 + 2119896) Γ (minus119904 + 2 + 2119896)

Γ (2k + 2)

(79)

a representation valid for all values of 119904 that is reminiscentof the following similar sum recently obtained by Tyagi andHolm ([31 Equation (35)])

120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1

(2 minus 2119904minus2119896) 120577 (minus119904 + 1 + 2119896) Γ (minus119904 + 1 + 2119896)

Γ (2119896 + 2)

120590 gt 0

(80)

(convergent by Gaussrsquo test) Equation (79) can also be rewrit-ten in the form

120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)

by replacing 119904 rarr 1 minus 119904 and applying (13)

With reference to (7) express the numerator 119888119900119904119894119899119890 factoras a difference of exponentials each of which in turn is writtenas a convergent power series in 120587(119888 + 119894119905) and recognize thatthe resulting series can be interchanged with the integral andeach term identified using (9) to find

120577 (119904) = minusradic120587infin

sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)

A more interesting variation of (82) can be obtained byapplying (3) and explicitly identifying the 119896 = 0 term of thesum giving

120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896

(1 minus 21minus119904+2119896) Γ (2119896 + 1) 120577 (119904 minus 2119896)

1 minus 2minus119904

(83)

a convergent representation if 120590 gt 1 (by Gaussrsquo test) Animmediate consequence is the identification

120577 (2) = minus1205872

3120577 (0) (84)

because only the 119896 = 1 term in the sum contributes when119904 = 2 Additionally when 119904 = 2119899 (83) terminates at 119899terms becoming a recursion for 120577(2119899) in terms of 120577(2119899 minus2119896) 119896 = 1 119899mdashsee Section 8 for further discussion Ifthe functional equation (13) is used to extend its region ofconvergence (83) becomes

120577 (119904) =1

2(120587119904

infin

sum119896=1

120577 (1 minus 119904 + 2119896) (2119904minus2119896minus1 minus 1)

Γ (2119896 + 1) Γ (119904 minus 2119896)

times ((2minus119904

minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)

which augments similar series appearing in [10 Sections 42and 43] and corresponds to a combination of entries in [28Equations (5463) and (5464)]

52 By Splitting

521 The Lower Range After applying (13) the integral in(42) is frequently (eg [11]) and conveniently split in twodefining

120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)


for arbitrary 120596 Similar to the derivation of (53) afterapplication of (13) we have

120577minus(119904) = minus2radic120587

infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)

which can be rewritten using (51) as

120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)

Substitute (1) in (90) interchange the order of summationand identify the resulting series to obtain

120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)

a convergent representation that is valid for all 119904Other representations are easily obtained by applying any

of the standard hypergeometric linear transforms as in (118)mdashsee Section 6 One such leads to

120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)

a rapidly converging representation for small values of 120596valid for all 119904 The inner sum of (92) can be written in severalways By expanding the denominator as a series in 1(11990521198982)with 119903 lt 2119901 minus 1 and interchanging the two series we find thegeneral form

120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0

Γ (119901 + 119898) (minus1)119898119905minus2119898minus2119901120577 (2119898 + 2119901 minus 119903)

Γ (119901) Γ (119898 + 1)

(93)

where the right-hand side continues the left if |119905| gt 1 and119903 minus 2119901 ge minus1 Alternatively [28 Equation (6132)] identifies

infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)

so for 119901 = 119896+1 and 119903 = 0 in (93) each term of the inner series(92) can be obtained analytically by taking the 119896th derivative

of the right-hand side with respect to the variable 1199102 at 119910 = 1leading to a rational polynomial in coth(120596) and sinh(120596) Thefirst few terms are given in Table 1 for two choices of 120596

Application of (93) to (92) leads to a less rapidly converg-ing representation that can be expressed in terms of morefamiliar functions

120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)

A second transform of the intermediate hypergeometricfunction yields another useful result

120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)

Both (95) and (96) can be reordered along a diagonal ofthe double sum (suminfin

119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus

119898)) allowing one of the resulting (terminating) series to beevaluated in closed form In either case both eventually give

120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)

reducing to (49) in the case 120596 = 120587 taking (13) intoconsideration Series of the form (97) have been studiedextensively [10 11] when 119904 = 119899 for special values of 120596 Theserepresentations will find application in Section 8

522TheUpper Range Rewrite (87) in exponential form andexpand the factor (1 minus eminus2119905)minus2 in a convergent power seriesinterchange the sum and integral and recognize that theresulting integral is a generalized exponential integral 119864

119904(119911)

usually defined in terms of incomplete Gamma functions([1 35] Section 819) as follows

119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)

giving a rapidly converging series representation

120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)


Table 1 The first few values of 120581(0 120581 + 1 12058721205962) in (93)

119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296

A useful variant of (99) is obtained by manipulating the well-known recurrence relation ([1 Equation 81912]) for119864

119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)

Apply (100) repeatedly119873 times to (99) and find

120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)

in terms of logarithms and polylogarithms (Li119896(119911)) a result

that loses numerical accuracy as119873 increases but which alsousefully terminates if 119904 = 119899119873 ge 119899 ge 0 This is discussed inmore detail in Section 8

Another useful representation emerges from (99) bywriting the function 119864

119904(2119896120596) in terms of confluent hyper-

geometric functions From [35 Equation (26b)] and [36Equation (9103)] we find

120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)

where the series on the right-hand side converges absolutely if120590 gt 1 independent of 120596 Comparison of (99) (88) and (102)identifies

120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)

which can be interpreted as a transformation of sums or ananalytic continuation for the various guises of 120577

minus(119904) given in

Section 521 if both or either of the two sides converge forsome choice of the variables 119904 andor 120596 respectively The factthat a term containing 120577(119904)naturally appears in the expression(102) for 120577

+(119904) suggests that there will be a severe cancellation

of digits between the two terms 120577minus(119904) and 120577

+(119904) if (88) is used

to attempt a numerical evaluation of 120577(119904) as has been reportedelsewhere [37] Essentially (88) is numerically useless (withcommon choices of arithmetic precision) for any choices of120588 gt 10 even if 120596 is carefully chosen in an attempt to balancethe cancellation of digitsHowever with 20 digits of precisionusing 20 terms in the series it is possible to find the firstzero of 120577(12 + 119894120588) correct to 10 digits deteriorating rapidlythereafter as 120588 increases For any choice of120596 the convergenceproperties of the two sums representing 120577

minus(119904) and 120577

+(119904) anti-

correlate as can be seen from Table 1 This can also be seenby rewriting (102) to yield the combined representation

21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)

In (104) the first inner term (119864minus119904(2119896120596)) vanishes like

exp(minus2119896120596)(2119896120596) whereas the second term (11198651(119904 + 1))

vanishes like 1119896(1+119904) minus exp(minus2119896120596)(2119896120596) so for large valuesof 119896 cancellation of digits is bound to occur (In factthe coefficients of both asymptotic series prefaced by theexponential terms cancel exactly term by term)

It should be noted that (102) is representative of a family ofrelated transformations For example although setting 119904 = 0in (102) does not lead to a new result operating first with 120597120597119904followed by setting 119904 = 0 leads to a transformation of a relatedsum

infin

sum119896=1

1198641(2119896120596) = minus

120596 arctanh (radicminus1205962120587)

120587radicminus1205962

minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)

where the first term corresponding to 119896 = 1 has been isolatedsince it contains the singularity that occurs due to divergenceof the left-hand sum if 120596 = 119894120587 (In (86) and (87) 120596 isin Cis permitted by a fundamental theorem of integral calculus)Similarly operating with 120597120597119904 on (102) at 119904 = 1 leads to theidentificationinfin

sum119896=1

log(1 + 1

12058721198962) = minus1 minus log (2) + log (1198902 minus 1) (106)


(which can alternatively be obtained by writing the infinitesum as a product) A second application of 120597120597119904 on (102)again at 119904 = 1 using (106) leads to the following transfor-mation of sums

1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12

minus 1 minus 120574 (1) minusinfin

sum119896=1

1198641(2119896)

119896

(107)

This transformation can be further reduced by applying anintegral representation given in [1 Section (62)] leading to

infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1

log (1 minus 119890minus2119905)

119905119889119905 (108)

Correcting a misprinted expression given in ([38 Equation(48)] [10 11] Equation 33(43)) which should read

infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0

log (120587radicV csc (120587radicV))119889V

V (109)

followed by a change of variables eventually yields therepresentationinfin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)

Substituting (108) and (110) into (107) followed by somesimplification finally gives

int1

0

log((1 minus 119890minus2V) (1198902V minus 1)

2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)

a result that appears to be new

53 Another Interpretation These expressions for thesplit representation can be interpreted differently Forthe moment let 120590 lt minus3 and insert the contour integralrepresentation [35 Equation (26a)] for 119864

minus119904(119911) into

the converging series representation (99) noting thatcontributions from infinity vanish giving

120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)

where the contour is defined by the real parameter 119888 lt minus2With this caveat the series in (112) converges allowing theintegration and summation to be interchanged leading to theidentification

120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin


Γ (minus119905) (2120596)119905120577 (minus1 minus 119905)

119904 + 1 + 119905119889119905 (113)

The contour may now be deformed to enclose the singularityat 119905 = minus119904 minus 1 translated to the right to pick up residues ofΓ(minus119905) at 119905 = 119899 119899 = 0 1 and the residue of 120577(minus1 minus 119905) at 119905 =minus2 After evaluation each of the residues so obtained cancelsone of the terms in the series expression (90) for 120577

minus(119904) leaving

(after a change of integration variables)

Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)

the standard Cauchy representation of the left-hand sidevalid for all 119904 where the contour encloses the simple pole at119905 = 119904

6 Integration by Parts

Integration by parts applied to previous results yields otherrepresentations

From (17) we obtain (known to theMaple computer codebut only in the form of a Mellin transform)

120577 (119904) = minus1

Γ (119904 minus 1)intinfin

0

119905119904minus2 log (1 minus exp (minus119905)) 119889119905 120590 gt 1

(115)

and from (64) we find

120577 (119904) =1

2minus

119904

120587

times intinfin

0

log (sinh (120587119905)) cos ((s + 1) arctan (119905))(1 + 1199052)

(1+119904)2

119889119905

120590 gt 1

(116)

as well as the following intriguing result (discovered by theMaple computer code)

120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905

minus 119904intinfin

0

119905 cosh (120587119905) cos (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)

This result has the interesting property that it is the onlyintegral representation of which I am aware in which theindependent variable 119904 does not appear as an exponentManyvariations of (117) can be obtained through the use of any


of the transformations of the hypergeometric function forexample

21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)

which choice happens to restore the variable 119904 to its usual roleas an exponent

7 Conversion to Integral Representations

From (79) identify the product 120577(sdot sdot sdot)Γ(sdot sdot sdot) in the form of anintegral representation from one of the primary definitionsof 120577(119904) ([13 Equation (9511)]) and interchange the series andsum The sums can now be explicitly evaluated leaving anintegral representation valid over an important range of 119904

120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0

119905minus119904 (119904 minus (2119904 sinh (1199052) 119905) minus 1 + cosh (1199052))119890119905 minus 1

119889119905

120590 lt 2

(119)

Each of the terms in (119) corresponds to known integrals([13 Equations (35521) and (83122)]) and if the identi-fication is taken to completion (119) reduces to the identity120577(119904) = 120577(119904)with the help of (13) Other useful representationsare immediately available by evaluating the integrals in (119)corresponding to pairs of terms in the numerator giving

120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0

119905minus119904 (minus1 + cosh (1199052))119890119905 minus 1

119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V

sinh (V)minus 1) 119890

minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899

Several of the results obtained in previous sections reduce tohelpful representations when 119904 = 119899

81 Finite Series Representations From (79) we find for evenvalues of the argument

120577 (2119899) = minus1

2(119899

sum119896=1

((minus1)1198961205872119896 (2119899 minus 2119896 minus 1) 120577 (2119899 minus 2119896)

Γ (2119896 + 2))

times (2minus2 119899

+ 119899 minus 1)minus1

)

(122)

from (83) we find

120577 (2119899) =1

2 (2minus2119899 minus 1)

times119899

sum119896=1

(minus1205872)119896

(1 minus 21minus2119899+2119896) 120577 (2119899 minus 2119896)

Γ (2119896 + 1)

(123)

and based on (80) we find [39] a similar but independentresult

120577 (2119899) = 22minus2119899


times119899

sum119896=1

((1 minus 22119899minus2119896minus1) (2120587)2119896120577 (2119899 minus 2119896) (minus1)119896

Γ (2119896 + 2))

(124)

three of an infinite number of such recursive relationships[40] The latter two may be identified with known results byreversing the sums giving

120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899

21minus2119899 minus 2(119899minus1

sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)

equivalent to [11 Equation 44(11)] and

120577 (2119899) =2 (minus1205872)

119899

1 minus 21minus2119899(119899minus1

sum119896=0

(1 minus 22119896minus1) 120577 (2119896) (minus1)119896

(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)

equivalent to [11 Equation 44(9)]

82 Novel Series Representations By adding and subtract-ing the first 119899 minus 1 terms in the series representation of


the hypergeometric function in (91) we find ([36 Chapter9]) for the case 119904 = 2119899 + 2 (119899 ge 0)

21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)

Substituting (128) or (129) in (88) (91) and (101) gives seriesrepresentations for 120577(2119899) and 120577(2119899+1) respectively for whichI find no record in the literature with particular reference toworks involving Barnes double Gamma function (eg [30])For example employing (128) in the case 119899 = 0 we obtainwhat appears to be a new representation (or the evaluation ofa new sum)

120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)

The series (130) (compare with ([28 Equation (4211)])) isabsolutely convergent by Gaussrsquo test for arbitrary values of 120596and reduces to an identity in the limiting case 120596 rarr 0

83 Connection with Multiple Gamma Function From Sec-tion 52 with 120596 = 1 we have after some simplification

120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2

) minus 2 log (1198902 minus 1)

(131)

Comparison of the previous infinite series with [10 Equation34(464)] using 119911 = 119894120587 identifies

log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)

minus4 log (1198902 minus 1) minus1205872

3)

= 0232662175652953119894

(132)

where119866 is Barnes doubleGamma function From [10 Section13] unnumbered equation following 26 we also find

log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)

thereby recovering (130) after representing the arctan func-tion in (130) as a logarithm (with 120596 = 1)

9 The Critical Strip

In (58) and (59) with 0 le 120590 le 1 let

119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)

Then after adding and subtracting we find a composition ofsymmetric and antisymmetric integrals about the critical line120590 = 12

120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0

ℎ (119905) cos (120588 ln (119905)) sinh((12minus 120590) ln (119905)) 119889119905

+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0

ℎ (119905) sin (120588 ln (119905)) cosh ((12minus 120590) ln (119905)) 119889119905

(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0

ℎ (119905) cos (120588 ln (119905)) cosh ((12minus 120590) ln (119905)) 119889119905

minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0

ℎ (119905) sin (120588 ln (119905)) sinh((12minus 120590) ln (119905)) 119889119905

(138)

Let

C (120588) = intinfin

0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)

For 120577(119904) = 0 on the critical line 120590 = 12 both (137) and(138) can be simultaneously satisfied when both 119862(120588) = 0 and119878(120588) = 0 only if

1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)

a true relationship that can be verified by direct computationSet 120590 = 12 in (137) and (138) and find

120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)

Equation (141) can be used to obtain a relationship betweenthe real and imaginary components of 120577(12 + 119894120588) Let

120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)

Then by breaking 119891(12 + 119894120588) and 119892(12 + 119894120588) into their realand imaginary parts (141) reduces to

120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862

119898(120588) sin (120588 ln (2120587)) minus radic120587

(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)

in terms of the real and imaginary parts of Γ(12 + 119894120588) Thisis a known result [41] that depends only on the functionalequation (13) (See the Appendix for further discussion onthis point) In [41] it was shown that the numerator anddenominator on the right-hand side of (11) can never vanishsimultaneously so this result defines the half-zeros (onlyone of 120577

119877(12 + 119894120588) or 120577

119868(12 + 119894120588) vanishes) (and usually

brackets the full-zeros) of 120577(12 + 119894120588) according to whetherthe numerator or denominator vanishes This was studied in[41] The full-zeros (120577

119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of

120577(12 + 119894120588) occur when the following condition

119878 (1205880) = 0

119862 (1205880) = 0

(146)

is simultaneously satisfied for some value(s) of 120588 = 1205880 In that

case the defining equation (143) becomes

120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))

+119862119898(120588) sin (120588 ln (2120587)) minus radic120587)

minus1

(147)

where the derivatives are taken with respect to the variable120588 and (147) is evaluated at 120588 = 120588

0 Many solutions to (146)

are known [42] so there is no doubt that these two functionscan simultaneously vanish at certain values of 120588 (A proof thatthis cannot occur in (137) and (138) when 120590 = 12 is left as anexercise for the reader)

A further interesting representation can be obtainedinside the critical strip by setting 119888 = 120590 in (9) and separating120578(119904) into its real and imaginary components


120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902

119890120588 arctan(119905) (cos (Ω (119905)) sin (120587120590) cosh (120587120590119905) minus sin (Ω (119905)) cos (120587120590) sinh (120587120590119905))cosh2 (120587120590119905) minus cos2 (120587120590)

119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902

119890120588 arctan(119905) (sin (Ω (119905)) sin (120587120590) cosh (120587120590119905) + cos (Ω (119905)) cos (120587120590) sinh (120587120590119905))cosh2 (120587120590119905) minus cos2 (120587120590)

119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)

In analogy to (137) and (138) the representations (148) and(149) include a term cos(120587120590) that vanishes on the critical line120590 = 12 both reducing to the equivalent of (26) in that case(I am speculating here that such forms contain informationabout the distribution andor existence of zeros of 120578(119904) andhence 120577(119904) inside the critical strip) The integrands in bothrepresentations vanish as exp(minus120587120590119905) as 119905 rarr infin and are finiteat 119905 = 0 most of the numerical contribution to the integralscomes from the region 0 lt 119905 cong 120588 Numerical evaluation inthe range 0 le 119905 can be acclerated by writing

e120588 arctan(119905) 997888rarr e120588(arctan(119905)minus1205872) (151)

and extracting a factor e1205881205872 outside the integral in that rangeIt remains to be shown how the remaining integral conspiresto cancel this factor in the asymptotic range 120588 rarr infinFinally (148) is invariant and (149) changes sign under thetransformations 120588 rarr minus120588 and 119905 rarr minus119905 demonstrating that120578(119904) is self-conjugate

10 Comments

A number of related results that are not easily categorizedand which either do not appear in the reference literature orappear incorrectly should be noted

Comparison of (49) and (54) suggests the identificationV = 119894 Such sums and integrals have been studied [30 43] forthe case 119904 = minus119899 in terms of Barnes double Gamma function119866(119911) and Clausenrsquos function defined by

Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)

with the (tacit) restriction that 119911 isin R That is not thecase here Thus straightforward application of that analysiswill sometimes lead to incorrect results For example [30Equation (43)] gives (with 119899 = 1)

120587int119905

0

V119899cot (120587V) 119889V = 119905 log (2120587) + log(119866 (1 minus 119905)

119866 (1 + 119905)) (153)

further ([43 Equation (13)]) identifying

log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)

To allow for the possibility that 119905 isin C this should instead read

log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)

which is valid for all 119905 in terms of Bernoulli polynomials119861119899(119905) and polylogarithm functions However similar results

are given in the equivalent general form in [20 Equation(512)] Interestingly both the Maple and Mathematica com-puter codes somehow (neither [44] nor references containedtherein both of which study such integrals list the computerresults which are likely based on [20 Equation (512)])manage to evaluate the integral in (153) in terms similar to(155) for a variety of values of 119899 provided that the user restricts|119905| lt 1 although the right-hand side provides the analyticcontinuation of the left when that restriction is liftedThis hasapplication in the evaluation of 120577(3) according to the methodof Section 83 where using the methods of [10 Chapter 3]we identify

infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)

a generalization of integrals studied in [44] From the com-puter evaluation of the integral (156) eventually yields

infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1

2minus log (1 minus 119890

2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)

as a new addition to the lists found in [10 Section (34)]and [11 Sections (33) and (34)] and a generalization of (97)when 119904 = 3


Appendix

In [41] an attempt was made to show that the functionalequation (13) and a minimal number of analytic propertiesof 120577(119904) might suffice to show that 120577(119904) = 0 when 120590 = 12 Thiswas motivated by a comment attributed to H Hamburger([8 page 50]) to the effect that the functional equation for120577(119904) perhaps characterizes it completely (An unequivocalequivalent claim can also be found in [15] As demonstratedby the counterexample given here and in [45] this claimis false) The purported demonstration contradicts a resultof Gauthier and Zeron [45] who show that infinitely manyfunctions exist arbitrarily close to 120577(119904) that satisfy (13) andthe same analytic properties that are invoked in [41] butwhich vanish when 120590 = 12 (To quote Prof Gauthier ldquoWeprove the existence of functions approximating the Riemannzeta-function satisfying the Riemann functional equationand violating the analogue of the Riemann hypothesisrdquo(private communication))

In fact an example of such a function is easily found Let

] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))

times sin (120587 (119904 minus 119904lowast

0)) sin (120587 (119904 + 119904

lowast

0))

(A1)

where 119904lowast0is the complex conjugate of 119904

0 1199040

= 1205900+ 1198941205880is

arbitrary complex and 119862 given by

119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)

is a constant chosen such that ](1) = 1 Then

120577] (119904) equiv ] (119904) 120577 (119904) (A3)

satisfies analyticity conditions (1ndash3) of [45] (which includethe functional equation (13)) Additionally 120577](119904) has zeros atthe zeros of 120577(119904) as well as complex zeros at 119904 = plusmn (119899+ 119904

0) and

119904 = plusmn (119899 + 119904lowast0) and a pole at 119904 = 1 with residue equal to unity

where 119899 = 0 plusmn1 plusmn2 Thus the last few paragraphs of Section 2 of [41] beginning

with the words ldquoThere are two cases where (29) fails rdquocannot be valid However the remainder of that paperparticularly those sections dealing with the properties of120577(12 + 119894120588) as referenced here in Section 9 are valid andcorrect I am grateful to Professor Gauthier for discussionson this topic

Acknowledgments

The author wishes to thank Vinicius Anghel and LarryGlasser for commenting on a draft copy of this paper LeslieM Saunders obituary may be found in the journal PhysicsToday issue of Dec 1972 page 63 httpwwwphysicstodayorgresource1phtoadv25i12p63 s2bypassSSO=1

References

[1] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress Cambridge UK 2010 httpdlmfnistgov255

[2] M Abramowitz and I Stegun Handbook of MathematicalFunctions Dover New York NY USA 1964

[3] T M Apostol Introduction to Analytic Number TheorySpringer New York NY USA 2010

[4] F W J Olver Asymptotics and Special Functions AcademicPress NewYorkNYUSA 1974 Computer Science andAppliedMathematics

[5] E C Titchmarsh The Theory of Functions Oxford UniversityPress New York NY USA 2nd edition 1949

[6] E TWhittaker andG NWatsonACourse ofModern AnalysisCambridge Mathematical Library Cambridge University PressCambridge UK 1963

[7] H M Edwards Riemannrsquos Zeta Function Academic Press NewYork NY USA 1974

[8] A Ivic The Riemann Zeta-Function Theory and ApplicationsDover New York NY USA 1985

[9] S J Patterson An introduction to the Theory of the RiemannZeta-Function vol 14 of Cambridge Studies in Advanced Math-ematics Cambridge University Press Cambridge UK 1988

[10] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic Dordrecht The Nether-lands 2001

[11] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier New York NY USA2012

[12] A ErdelyiWMagnus FOberhettinger and FG Tricomi EdsHigher Transcendental Functions vol 1ndash3 McGraw-Hill NewYork NY USA 1953

[13] I S Gradshteyn and I M Ryzhik Tables of Integrals Series andProducts Academic Press New York NY USA 1980

[14] A P Prudnikov Yu A Brychkov and O I Marichev Integralsand Series Volume 3 More Special Functions Gordon andBreach Science New York NY USA 1990

[15] Encyclopedia of mathematics httpwwwencyclopediaofmathorgindexphpZeta-function

[16] httpenwikipediaorgwikiRiemann zeta function[17] httpenwikipediaorgwikiDirichlet eta function[18] Mathworld httpmathworldwolframcomHankelFunction

html[19] Mathworld httpmathworldwolframcomRiemannZeta-

Functionhtml[20] H M Srivastava M L Glasser and V S Adamchik ldquoSome

definite integrals associated with the Riemann zeta functionrdquoZeitschrift fur Analysis und ihre Anwendung vol 19 no 3 pp831ndash846 2000

[21] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[22] R G NewtonThe Complex J-Plane The Mathematical PhysicsMonograph Series W A Benjamin New York NY USA 1964

[23] J L W V Jensen LrsquoIntermediaire des mathematiciens vol 21895

[24] E LindelofLeCalculDes Residus et ses Applications a LaTheorieDes Fonctions Gauthier-Villars Paris France 1905

[25] S N M Ruijsenaars ldquoSpecial functions defined by analyticdifference equationsrdquo in Special Functions 2000 Current Per-spective and Future Directions (Tempe AZ) J Bustoz M Ismailand S Suslov Eds vol 30 of NATO Science Series pp 281ndash333Kluwer Academic Dordrecht The Netherlands 2001


[26] B Riemann ldquoUber die Anzahl der Primzahlen unter einergegebenen Grosserdquo 1859

[27] T Amdeberhan M L Glasser M C Jones V Moll RPosey andD Varela ldquoTheCauchy-Schlomilch transformationrdquohttparxivorgabs10042445

[28] E R Hansen Table of Series and Products Prentice Hall NewYork NY USA 1975

[29] R E Crandall Topics in Advanced Scientific ComputationSpringer New York NY USA 1996

[30] J Choi H M Srivastava and V S Adamchick ldquoMultiplegamma and related functionsrdquo Applied Mathematics and Com-putation vol 134 no 2-3 pp 515ndash533 2003

[31] S Tyagi and C Holm ldquoA new integral representation for theRiemann zeta functionrdquo httparxivorgabsmath-ph0703073

[32] N G de Bruijn ldquoMathematicardquo Zutphen vol B5 pp 170ndash1801937

[33] R B Paris and D Kaminski Asymptotics and Mellin-BarnesIntegrals vol 85 of Encyclopedia of Mathematics and Its Appli-cations Cambridge University Press Cambridge UK 2001

[34] Y L LukeThe Special Functions andTheir Approximations vol1 Academic Press New York NY USA 1969

[35] M S Milgram ldquoThe generalized integro-exponential functionrdquoMathematics of Computation vol 44 no 170 pp 443ndash458 1985

[36] N N Lebedev Special Functions and Their Applications Pren-tice Hall New York NY USA 1965

[37] TMDunster ldquoUniform asymptotic approximations for incom-plete Riemann zeta functionsrdquo Journal of Computational andApplied Mathematics vol 190 no 1-2 pp 339ndash353 2006

[38] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[39] M S Milgram ldquoNotes on a paper of Tyagi and Holm lsquoanew integral representation for the Riemann Zeta functionrsquordquohttparxivorgabs07100037v1

[40] E Y Deeba andDM Rodriguez ldquoStirlingrsquos series and Bernoullinumbersrdquo American Mathematical Monthly vol 98 no 5 pp423ndash426 1991

[41] M SMilgram ldquoNotes on the zeros of Riemannrsquos zeta Functionrdquohttparxivorgabs09111332

[42] A M Odlyzko ldquoThe 1022-nd zero of the Riemann zeta func-tionrdquo in Dynamical Spectral and Arithmetic Zeta Functions(San Antonio TX 1999) vol 290 of Contemporary MathematicsSeries pp 139ndash144 American Mathematical Society Provi-dence RI USA 2001

[43] V S Adamchik ldquoContributions to the theory of the barnes func-tion computer physics communicationrdquo httparxivorgabsmath0308086

[44] DCvijovic andHM Srivastava ldquoEvaluations of some classes ofthe trigonometric moment integralsrdquo Journal of MathematicalAnalysis and Applications vol 351 no 1 pp 244ndash256 2009

[45] P M Gauthier and E S Zeron ldquoSmall perturbations ofthe Riemann zeta function and their zerosrdquo ComputationalMethods and Function Theory vol 4 no 1 pp 143ndash150 2004httpwwwheldermann-verlagdecmfcmf04cmf0411pdf

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The Riemann Zeta function 120577(119904) and the Dirichlet alter-nating zeta function 120578(119904) are well known and defined by(convergent) series representations

120577 (119904) equivinfin

sum119896=1

119896minus119904

120590 gt 1 (1)

120578 (119904) equiv minusinfin

sum119896=1

(minus1)119896

119896minus119904

120590 gt 0 (2)

with

120578 (119904) = (1 minus 21minus119904

) 120577 (119904) (3)

It is easily found by an elementary application of the residuetheorem that the following reproduces (1)

120577 (119904) = 120587infin

sum119896=1

residue 119905minus119904119890119894120587119905

sin (120587119905)

100381610038161003816100381610038161003816100381610038161003816119905=119896 120590 gt 1 (4)

and the following reproduces (2)

120578 (119904) = 120587infin

sum119896=1

residue(119905minus119904cot (120587119905) 119890119894120587119905)10038161003816100381610038161003816119905=119896 120590 gt 0 (5)

Convert the result (4) into a contour integral enclosing thepositive integers on the 119905-axis in a clockwise direction andprovided that 120590 gt 1 so that contributions from infinityvanish the contour may then be opened such that it stretchesvertically in the complex 119905-plane giving

120577 (119904) =119894

2int119888+119894infin


119905minus119904cot (120587119905) 119889119905 0 lt 119888 lt 1 119888 isin R (6)

= minus1

2intinfin

minusinfin

(119888 + 119894119905)minus119904cot (120587 (119888 + 119894119905)) 119889119905 120590 gt 1 (7)

a result that cannot be found in any of the modern referenceworks cited (in particular [20]) It should be noted thatthis procedure is hardly novelmdashit is often employed in aneducational context (eg in the special case 119904 = 2) todemonstrate the utility of complex integration to evaluatespecial sums ([10 Section 41]) ([21 Lemma 21]) and formsthe basis for an entire branch of physics [22]

A similar application to (5) with some trivial trigonomet-ric simplification gives

120578 (119904) =minus119894

2int119888+119894infin


119905minus119904

sin (120587119905)119889119905 (8)

=1

2intinfin

minusinfin

(119888 + 119894119905)minus119904

sin (120587 (119888 + 119894119905))119889119905 (9)

Equation (9) reduces with 119888 = 12 to a Jensen result[23] presented (without proof) in 1895 and to the best ofmy knowledge reproduced again but only in the specialcase 119888 = 12 only once in the modern reference literature[17] In particular although Lindelof rsquos 1905 work [24] arrivesat a result equivalent to that obtained here attributing thetechnique to Cauchy (1827) it is performed in a general

context so that the specific results (7) and (9) never appearparticularly in that section of the work devoted to 120577(119904) Inmodern notation for ameromorphic function119891(119911) Lindelofwrites ([24 Equation III(4)]) in general

infin

sum]=minusinfin

119891 (]) = minus1

2120587119894∮120587 cot (120587119911) 119891 (119911) 119889119911 (10)

where the contour of integration encloses all the singularitiesof the integrand but never specifically identifies 119891(119911) =119911minus119904 Much later after the integral has been split into twoby the invocation of identities for cot(120587119911) he does makethis identification ([24 Equation 4III (6)]) arriving (with(3)) at ((26)mdashsee below) a now-well-known result therebybypassing the contour integral representation (7) In a similarvein Olver ([4 page 290]) reproduces (10) employing itto evaluate remainder terms the Abel-Plana formula and aJensen result but again never writes (7) explicitly Perhaps theomission of an explicit statement of (7) and (9) from ChapterIV of Lindelof [24] is why these forms seem to have vanishedfrom the historical record although citations to Jensen andLindelof abound inmodern summaries (Recently Srivastavaand Choi [11 pages 169ndash172] have reproduced Lindelof rsquosanalysis also failing to arrive explicitly at (7) although theydo write a general form of (8) ([11 Section 23 Equation(38)]) it is studied only for the case 119888 = 12) (Note added inproof after a draft copy of this paper was posted I was madeaware that Ruijsenaars has previously obtained a generalizedform of (7) for the Hurwitz zeta function ([25 Equation(422)]) He also notes (private communication) that he couldnot find prior references to his result in the literature)

At this point it is worthwhile to quote several relatedintegral representations that define some of the relevantand important functions of complex analysis starting withRiemannrsquos original 1859 contour integral representation of thezeta function [26]

120577 (119904) =Γ (1 minus 119904)

2120587119894∮

119905119904minus1

119890minus119905 minus 1119889119905 (11)

where the contour of integration encloses the negative 119905-axislooping from 119905 = minusinfinminus 1198940 to 119905 = minusinfin+ 1198940 enclosing the point119905 = 0

The equivalence of (11) and (1) is well described in texts(eg [11]) and is obtained by reducing three components ofthe contour in (11) A different analysis is possible however(eg [7 Section 16])mdashopen and translate the contour in (11)such that it lies vertically to the right of the origin in thecomplex 119905-plane (and thereby vanishes) with 120590 lt 0 andevaluate the residues of each of the poles of the integrandlying at 119905 = plusmn 2119899120587119894 to find

120577 (119904) =minus119894

2120587Γ (1 minus 119904) (2120587)

119904

times (exp (1198941205871199042) minus exp (minus119894120587119904

2))infin

sum119896=1

119896119904minus1

(12)



120577 (1 minus 119904) =2 cos (1205871199042) Γ (119904)

(2120587)119904120577 (119904) (13)



120577 (119904) =120587


minusinfin

(12 + 119894119905)1minus119904

cosh2 (120587119905)119889119905 120590 gt 0 (14)


120577 (119904) =2119904minus1


0

119905119904

cosh2 (119905)119889119905 120590 gt minus1 (15)



([12 Equation 112(6)])

120577 (119904) =1


0

119905119904minus1

sinh (119905)119889119905 120590 gt 1 (16)

([12 Equation 112(4)])

120577 (119904) =1

Γ (119904)intinfin

0

119905119904minus1

exp (119905) minus 1119889119905 120590 gt 1 (17)


120577 (119904) =1

4

(2119886)(119904+1)


0

119905119904

sinh2 (119886119905)119889119905 120590 gt 1 (18)


1

Γ (119904)=

1

2120587119894intinfin

minusinfin

119890(119888+119894119905)

(119888 + 119894119905)119904119889119905 (19)


3 Simple Reductions


(minus(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

) 119894 = 2(1198882

+ 1199052

)(1199042)

times sin(119904 arctan( 119905119888))

(20)

(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

= 2(1198882

+ 1199052

)(1199042)

cos(119904 arctan( 119905119888))

(21)



0



times ((1198882

+ 1199052

)1199042


119889119905

120590 gt 1

(22)

120578 (119904) = intinfin

0



times ((1198882

+ 1199052

)1199042


119889119905

(23)


120577 (119904) = 2119904

intinfin

0

sinh (120587119905) sin (119904 arctan (2119905))(1 + 41199052)

1199042 cosh (120587119905)119889119905 120590 gt 1 (24)


120577 (119904) =2(119904minus1)

119904 minus 1minus 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890minus120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 (25)


120578 (119904) = 2119904minus1

intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042 cosh (1205871199052)119889119905 (26)


120577 (119904) =2(119904minus1)

1 minus 119904+ 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 120590 gt 1

(27)




0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (28)


120577 (119904) =sin (1199041205872)(21minus119904 minus 1)

intinfin

0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (29)


120577 (119904) = 3119904minus1

intinfin

0

(2 sin (119904 arctan (119905)) sinh(21205871199053

)


times ((1 + 1199052

)1199042

(2 cosh (21205871199053

) + 1))minus1

119889119905

120590 gt 1

(30)


120578 (119904) =minus119894

2int119904+119894infin


119905minus119904

sin (120587119905)119889119905 (31)

=1

2intinfin

minusinfin

(119904 + 119894119905)minus119904

sin (120587 (119904 + 119894119905))119889119905 (32)





120597120578 (119904)

120597119888= 0 119888 =119873 (33)


120578 (119904 + 1) = minus120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904 cos (120587 (119888 + 119894119905))

sin2 (120587 (119888 + 119894119905))119889119905

0 le 119888 lt 1

(34)


120577 (119904 + 1) =120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904

sin2 (120587 (119888 + 119894119905))119889119905 0 lt 119888 lt 1 (35)


120578 (119904 + 1) =2119904minus1120587

119904

times intinfin

0

(1 + 1199052

)minus1199042

sinh(1205871199052) sin (119904 arctan (119905))

times coshminus2 (1205871199052) 119889119905

(36)


120578 (119904 + 1) =2 (119904 minus 1) 120578 (119904)

119904+2119904minus1120587

119904

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(37)



119904 (1 minus 2minus119904)+

2119904minus1120587

119904 (1 minus 2minus119904)

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(38)



120577 (119904 + 1) =1205872119904minus1

119904intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042cosh2 (1205871199052)119889119905 (39)


120578 (119904) =120587 sin (1199041205872)


0

1199051minus119904 cosh (120587119905)sinh2 (120587119905)

119889119905 120590 lt 0 (40)


120578 (119904 + 1) = 0 1 +119894120587119903

2119904

times int120587

0

(1 plusmn 119903119894119890119894120579)minus119904

cosh (120587119903119890119894120579) 119890119894120579

sinh2 (120587119903119890119894120579)119889120579

minus119903120587

119904intinfin

1

(1 + 1199032

1199052

)minus1199042

cos (119904 arctan (119903119905))


(41)


4 Special Cases


120577 (119904) = minus120587119904minus1

sin (1205871199042)119904 minus 1

intinfin

0

1199051minus119904

sinh2 (119905)119889119905 120590 lt 0 (42)


120577 (119904) = 1205770(119904) + 120577

1(119904) (43)

where

1205770(119904) =

119894

2∮119894

minus119894

119905minus119904cot (120587119905) 119889119905 (44)

1205771(119904) =

119894

2int119894

minus119894infin

119905minus119904cot (120587119905) 119889119905 + 119894

2int119894infin

119894

119905minus119904cot (120587119905) 119889119905 (45)


1205771(119904) = sin(120587119904

2)intinfin

1

119905minus119904 coth (120587119905) 119889119905 120590 gt 1 (46)

=sin (1205871199042)119904 minus 1

+ sin(1205871199042)intinfin

1

119905minus119904119890(minus120587119905)

sinh (120587119905)119889119905 (47)


119905 cot (120587119905) = minus2

120587

infin

sum119896=0

1199052119896

120577 (2119896) |119905| lt 1 (48)

giving

1205770(119904) = minus

2

120587sin(120587119904

2)infin

sum119896=0

(minus1)119896120577 (2119896)

2119896 minus 119904 (49)


0(119904) = 120577(119904) and (47) vanishes


1

sin2 (120587V)=

2

1205872

infin

sum119896=0

(2119896 minus 1) 120577 (2119896) V2119896minus2

|V| lt 1 (50)



120577 (2119896) = minus1

2

(minus1)119896(2120587)21198961198612119896

Γ (2119896 + 1) (51)


120577 (119904) = minus1

2

119894120587

(119904 minus 1)int119894

minus119894

1199051minus119904

sin2 (120587119905)119889119905

minus120587 sin (1205871199042)


1

1199051minus119904

sinh2 (120587119905)119889119905

(52)

becomes (formally)

120577 (119904) = minussin (1205871199042)119904 minus 1

times (2

120587

infin

sum119896=0

(minus1)119896 (2119896 minus 1) 120577 (2119896)

2119896 minus 119904

+120587intinfin

1

1199051minus119904

sinh2 (120587119905)119889119905)

(53)




sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)


infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)


120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)


120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)



120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)


119873 (If 119888119873


120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)





119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1


0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)


120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)


120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120577 (1 minus 119904) =2 cos (1205871199042) Γ (119904)

(2120587)119904120577 (119904) (13)



120577 (119904) =120587


minusinfin

(12 + 119894119905)1minus119904

cosh2 (120587119905)119889119905 120590 gt 0 (14)


120577 (119904) =2119904minus1


0

119905119904

cosh2 (119905)119889119905 120590 gt minus1 (15)



([12 Equation 112(6)])

120577 (119904) =1


0

119905119904minus1

sinh (119905)119889119905 120590 gt 1 (16)

([12 Equation 112(4)])

120577 (119904) =1

Γ (119904)intinfin

0

119905119904minus1

exp (119905) minus 1119889119905 120590 gt 1 (17)


120577 (119904) =1

4

(2119886)(119904+1)


0

119905119904

sinh2 (119886119905)119889119905 120590 gt 1 (18)


1

Γ (119904)=

1

2120587119894intinfin

minusinfin

119890(119888+119894119905)

(119888 + 119894119905)119904119889119905 (19)


3 Simple Reductions


(minus(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

) 119894 = 2(1198882

+ 1199052

)(1199042)

times sin(119904 arctan( 119905119888))

(20)

(119888 + 119894119905)119904

+ (119888 minus 119894119905)119904

= 2(1198882

+ 1199052

)(1199042)

cos(119904 arctan( 119905119888))

(21)



0



times ((1198882

+ 1199052

)1199042


119889119905

120590 gt 1

(22)

120578 (119904) = intinfin

0



times ((1198882

+ 1199052

)1199042


119889119905

(23)


120577 (119904) = 2119904

intinfin

0

sinh (120587119905) sin (119904 arctan (2119905))(1 + 41199052)

1199042 cosh (120587119905)119889119905 120590 gt 1 (24)


120577 (119904) =2(119904minus1)

119904 minus 1minus 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890minus120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 (25)


120578 (119904) = 2119904minus1

intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042 cosh (1205871199052)119889119905 (26)


120577 (119904) =2(119904minus1)

1 minus 119904+ 2119904

intinfin

0

sin (119904 arctan (2119905)) 119890120587119905

(1 + 41199052)1199042 cosh (120587119905)

119889119905 120590 gt 1

(27)




0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (28)


120577 (119904) =sin (1199041205872)(21minus119904 minus 1)

intinfin

0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (29)


120577 (119904) = 3119904minus1

intinfin

0

(2 sin (119904 arctan (119905)) sinh(21205871199053

)


times ((1 + 1199052

)1199042

(2 cosh (21205871199053

) + 1))minus1

119889119905

120590 gt 1

(30)


120578 (119904) =minus119894

2int119904+119894infin


119905minus119904

sin (120587119905)119889119905 (31)

=1

2intinfin

minusinfin

(119904 + 119894119905)minus119904

sin (120587 (119904 + 119894119905))119889119905 (32)





120597120578 (119904)

120597119888= 0 119888 =119873 (33)


120578 (119904 + 1) = minus120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904 cos (120587 (119888 + 119894119905))

sin2 (120587 (119888 + 119894119905))119889119905

0 le 119888 lt 1

(34)


120577 (119904 + 1) =120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904

sin2 (120587 (119888 + 119894119905))119889119905 0 lt 119888 lt 1 (35)


120578 (119904 + 1) =2119904minus1120587

119904

times intinfin

0

(1 + 1199052

)minus1199042

sinh(1205871199052) sin (119904 arctan (119905))

times coshminus2 (1205871199052) 119889119905

(36)


120578 (119904 + 1) =2 (119904 minus 1) 120578 (119904)

119904+2119904minus1120587

119904

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(37)



119904 (1 minus 2minus119904)+

2119904minus1120587

119904 (1 minus 2minus119904)

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(38)



120577 (119904 + 1) =1205872119904minus1

119904intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042cosh2 (1205871199052)119889119905 (39)


120578 (119904) =120587 sin (1199041205872)


0

1199051minus119904 cosh (120587119905)sinh2 (120587119905)

119889119905 120590 lt 0 (40)


120578 (119904 + 1) = 0 1 +119894120587119903

2119904

times int120587

0

(1 plusmn 119903119894119890119894120579)minus119904

cosh (120587119903119890119894120579) 119890119894120579

sinh2 (120587119903119890119894120579)119889120579

minus119903120587

119904intinfin

1

(1 + 1199032

1199052

)minus1199042

cos (119904 arctan (119903119905))


(41)


4 Special Cases


120577 (119904) = minus120587119904minus1

sin (1205871199042)119904 minus 1

intinfin

0

1199051minus119904

sinh2 (119905)119889119905 120590 lt 0 (42)


120577 (119904) = 1205770(119904) + 120577

1(119904) (43)

where

1205770(119904) =

119894

2∮119894

minus119894

119905minus119904cot (120587119905) 119889119905 (44)

1205771(119904) =

119894

2int119894

minus119894infin

119905minus119904cot (120587119905) 119889119905 + 119894

2int119894infin

119894

119905minus119904cot (120587119905) 119889119905 (45)


1205771(119904) = sin(120587119904

2)intinfin

1

119905minus119904 coth (120587119905) 119889119905 120590 gt 1 (46)

=sin (1205871199042)119904 minus 1

+ sin(1205871199042)intinfin

1

119905minus119904119890(minus120587119905)

sinh (120587119905)119889119905 (47)


119905 cot (120587119905) = minus2

120587

infin

sum119896=0

1199052119896

120577 (2119896) |119905| lt 1 (48)

giving

1205770(119904) = minus

2

120587sin(120587119904

2)infin

sum119896=0

(minus1)119896120577 (2119896)

2119896 minus 119904 (49)


0(119904) = 120577(119904) and (47) vanishes


1

sin2 (120587V)=

2

1205872

infin

sum119896=0

(2119896 minus 1) 120577 (2119896) V2119896minus2

|V| lt 1 (50)



120577 (2119896) = minus1

2

(minus1)119896(2120587)21198961198612119896

Γ (2119896 + 1) (51)


120577 (119904) = minus1

2

119894120587

(119904 minus 1)int119894

minus119894

1199051minus119904

sin2 (120587119905)119889119905

minus120587 sin (1205871199042)


1

1199051minus119904

sinh2 (120587119905)119889119905

(52)

becomes (formally)

120577 (119904) = minussin (1205871199042)119904 minus 1

times (2

120587

infin

sum119896=0

(minus1)119896 (2119896 minus 1) 120577 (2119896)

2119896 minus 119904

+120587intinfin

1

1199051minus119904

sinh2 (120587119905)119889119905)

(53)




sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)


infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)


120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)


120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)



120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)


119873 (If 119888119873


120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)





119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1


0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)


120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)


120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (28)


120577 (119904) =sin (1199041205872)(21minus119904 minus 1)

intinfin

0

119905minus119904

sinh (120587119905)119889119905 120590 lt 0 (29)


120577 (119904) = 3119904minus1

intinfin

0

(2 sin (119904 arctan (119905)) sinh(21205871199053

)


times ((1 + 1199052

)1199042

(2 cosh (21205871199053

) + 1))minus1

119889119905

120590 gt 1

(30)


120578 (119904) =minus119894

2int119904+119894infin


119905minus119904

sin (120587119905)119889119905 (31)

=1

2intinfin

minusinfin

(119904 + 119894119905)minus119904

sin (120587 (119904 + 119894119905))119889119905 (32)





120597120578 (119904)

120597119888= 0 119888 =119873 (33)


120578 (119904 + 1) = minus120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904 cos (120587 (119888 + 119894119905))

sin2 (120587 (119888 + 119894119905))119889119905

0 le 119888 lt 1

(34)


120577 (119904 + 1) =120587

2119904intinfin

minusinfin

(119888 + 119894119905)minus119904

sin2 (120587 (119888 + 119894119905))119889119905 0 lt 119888 lt 1 (35)


120578 (119904 + 1) =2119904minus1120587

119904

times intinfin

0

(1 + 1199052

)minus1199042

sinh(1205871199052) sin (119904 arctan (119905))

times coshminus2 (1205871199052) 119889119905

(36)


120578 (119904 + 1) =2 (119904 minus 1) 120578 (119904)

119904+2119904minus1120587

119904

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(37)



119904 (1 minus 2minus119904)+

2119904minus1120587

119904 (1 minus 2minus119904)

times intinfin

0

cos (119904 arctan (119905)) (1 + 1199052

)minus1199042

sinh(1205871199052)

times coshminus2 (1205871199052) 119905 119889119905

(38)



120577 (119904 + 1) =1205872119904minus1

119904intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042cosh2 (1205871199052)119889119905 (39)


120578 (119904) =120587 sin (1199041205872)


0

1199051minus119904 cosh (120587119905)sinh2 (120587119905)

119889119905 120590 lt 0 (40)


120578 (119904 + 1) = 0 1 +119894120587119903

2119904

times int120587

0

(1 plusmn 119903119894119890119894120579)minus119904

cosh (120587119903119890119894120579) 119890119894120579

sinh2 (120587119903119890119894120579)119889120579

minus119903120587

119904intinfin

1

(1 + 1199032

1199052

)minus1199042

cos (119904 arctan (119903119905))


(41)


4 Special Cases


120577 (119904) = minus120587119904minus1

sin (1205871199042)119904 minus 1

intinfin

0

1199051minus119904

sinh2 (119905)119889119905 120590 lt 0 (42)


120577 (119904) = 1205770(119904) + 120577

1(119904) (43)

where

1205770(119904) =

119894

2∮119894

minus119894

119905minus119904cot (120587119905) 119889119905 (44)

1205771(119904) =

119894

2int119894

minus119894infin

119905minus119904cot (120587119905) 119889119905 + 119894

2int119894infin

119894

119905minus119904cot (120587119905) 119889119905 (45)


1205771(119904) = sin(120587119904

2)intinfin

1

119905minus119904 coth (120587119905) 119889119905 120590 gt 1 (46)

=sin (1205871199042)119904 minus 1

+ sin(1205871199042)intinfin

1

119905minus119904119890(minus120587119905)

sinh (120587119905)119889119905 (47)


119905 cot (120587119905) = minus2

120587

infin

sum119896=0

1199052119896

120577 (2119896) |119905| lt 1 (48)

giving

1205770(119904) = minus

2

120587sin(120587119904

2)infin

sum119896=0

(minus1)119896120577 (2119896)

2119896 minus 119904 (49)


0(119904) = 120577(119904) and (47) vanishes


1

sin2 (120587V)=

2

1205872

infin

sum119896=0

(2119896 minus 1) 120577 (2119896) V2119896minus2

|V| lt 1 (50)



120577 (2119896) = minus1

2

(minus1)119896(2120587)21198961198612119896

Γ (2119896 + 1) (51)


120577 (119904) = minus1

2

119894120587

(119904 minus 1)int119894

minus119894

1199051minus119904

sin2 (120587119905)119889119905

minus120587 sin (1205871199042)


1

1199051minus119904

sinh2 (120587119905)119889119905

(52)

becomes (formally)

120577 (119904) = minussin (1205871199042)119904 minus 1

times (2

120587

infin

sum119896=0

(minus1)119896 (2119896 minus 1) 120577 (2119896)

2119896 minus 119904

+120587intinfin

1

1199051minus119904

sinh2 (120587119905)119889119905)

(53)




sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)


infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)


120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)


120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)



120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)


119873 (If 119888119873


120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)





119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1


0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)


120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)


120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120577 (119904 + 1) =1205872119904minus1

119904intinfin

0

cos (119904 arctan (119905))(1 + 1199052)

1199042cosh2 (1205871199052)119889119905 (39)


120578 (119904) =120587 sin (1199041205872)


0

1199051minus119904 cosh (120587119905)sinh2 (120587119905)

119889119905 120590 lt 0 (40)


120578 (119904 + 1) = 0 1 +119894120587119903

2119904

times int120587

0

(1 plusmn 119903119894119890119894120579)minus119904

cosh (120587119903119890119894120579) 119890119894120579

sinh2 (120587119903119890119894120579)119889120579

minus119903120587

119904intinfin

1

(1 + 1199032

1199052

)minus1199042

cos (119904 arctan (119903119905))


(41)


4 Special Cases


120577 (119904) = minus120587119904minus1

sin (1205871199042)119904 minus 1

intinfin

0

1199051minus119904

sinh2 (119905)119889119905 120590 lt 0 (42)


120577 (119904) = 1205770(119904) + 120577

1(119904) (43)

where

1205770(119904) =

119894

2∮119894

minus119894

119905minus119904cot (120587119905) 119889119905 (44)

1205771(119904) =

119894

2int119894

minus119894infin

119905minus119904cot (120587119905) 119889119905 + 119894

2int119894infin

119894

119905minus119904cot (120587119905) 119889119905 (45)


1205771(119904) = sin(120587119904

2)intinfin

1

119905minus119904 coth (120587119905) 119889119905 120590 gt 1 (46)

=sin (1205871199042)119904 minus 1

+ sin(1205871199042)intinfin

1

119905minus119904119890(minus120587119905)

sinh (120587119905)119889119905 (47)


119905 cot (120587119905) = minus2

120587

infin

sum119896=0

1199052119896

120577 (2119896) |119905| lt 1 (48)

giving

1205770(119904) = minus

2

120587sin(120587119904

2)infin

sum119896=0

(minus1)119896120577 (2119896)

2119896 minus 119904 (49)


0(119904) = 120577(119904) and (47) vanishes


1

sin2 (120587V)=

2

1205872

infin

sum119896=0

(2119896 minus 1) 120577 (2119896) V2119896minus2

|V| lt 1 (50)



120577 (2119896) = minus1

2

(minus1)119896(2120587)21198961198612119896

Γ (2119896 + 1) (51)


120577 (119904) = minus1

2

119894120587

(119904 minus 1)int119894

minus119894

1199051minus119904

sin2 (120587119905)119889119905

minus120587 sin (1205871199042)


1

1199051minus119904

sinh2 (120587119905)119889119905

(52)

becomes (formally)

120577 (119904) = minussin (1205871199042)119904 minus 1

times (2

120587

infin

sum119896=0

(minus1)119896 (2119896 minus 1) 120577 (2119896)

2119896 minus 119904

+120587intinfin

1

1199051minus119904

sinh2 (120587119905)119889119905)

(53)




sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)


infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)


120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)


120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)



120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)


119873 (If 119888119873


120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)





119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1


0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)


120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)


120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum119896=0

120577 (2119896) V2119896

2119896 minus 119904=

1

2119904+ int

V

0

119905minus119904

infin

sum119896=1

120577 (2119896) 1199052119896minus1

119889119905

(54)

=1

2119904+1

2V119904

intV

0

119905minus119904

(Ψ (1 + 119905) minus Ψ (1 minus 119905)) 119889119905

120590 lt 2

(55)


infin

sum119896=0

120577 (2119896) (2119896 minus 1) V2119896

2119896 minus 119904

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(Ψ1015840

(1 + 119905) + Ψ1015840

(1 minus 119905)) 119889119905

(56)

= minus1

2119904+1

2V119904

times intV

0

1199051minus119904

(1205872

(1 + cot2 (120587119905)) minus 1

1199052)119889119905

120590 lt 2

(57)


120577 (119904) = minus120587119904minus1 sin (1205871199042)


0

1199051minus119904

(1

sinh2119905minus

1

1199052)119889119905 (58)


120577 (119904) =2119904minus1


0

119905119904

(1

sinh2119905minus

1

1199052)119889119905 (59)



120577119873(119904) equiv

119873

sum119896=1

119896minus119904

120578119873(119904) equiv minus

119873

sum119896=1

(minus1)119896

119896minus119904

(60)

let 119888 rarr 119888119873119873 = 1 2 3 with

119873 le 119888119873lt 119873 + 1 (61)


119873 (If 119888119873


120577 (119904) = 120577119873(119904) minus

1

2119873119904120575119873119888119873

+119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904cot (120587t) 119889119905 120590 gt 1

(62)

and (9)

120578 (119904) = 120578119873(119904) +

(minus1)119873

2119873119904120575119873119888119873

minus119894

2int119888119873+119894infin

119888119873minus119894infin

119905minus119904

sin (120587119905)119889119905 (63)





119873= 119873 = 1 one finds

120577 (119904) =1

2+ intinfin

0

cosh (120587119905) sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 120590 gt 1 (64)

=119904 + 1


0

119890minus120587119905 sin (119904 arctan (119905))(1 + 1199052)

1199042 sinh (120587119905)119889119905 (65)

120578 (119904) =1

2+ intinfin

0

sin (119904 arctan (119905))sinh (120587119905) (1 + 1199052)

1199042

119889119905 (66)


120577 (119904) =2119904minus1119904

(2119904 minus 1) (119904 minus 1)+

2119904minus1

2119904 minus 1

times intinfin

0

sin (119904 arctan (119905)) (1 + 119890minus120587119905)

(1 + 1199052)1199042 sinh (120587119905)

119889119905

(67)


120577 (119904) =119904 minus 3


0

sin (119904 arctan (119905)) 119890120587119905

(1 + 1199052)1199042 sinh (120587119905)

119889119905 120590 gt 1

(68)


For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










For 119888119873



120577 (119904) minus119873

sum119896=1

1

119896119904=

1

2(1

2+ 119873)1minus119904

intinfin

0

sinh (120587119905 (1 + 2119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905 120590 gt 1 (69)

=(12 + 119873)

1minus119904


1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1 + 119890120587119905(1+2119873)

119889119905 (70)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904= (minus1)

119873

(1

2+ 119873)1minus119904

intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh (120587119905 (12 + 119873))

119889119905 (71)


119873= 12 + 119873


120577 (119904 + 1) minus119873

sum119896=1

1

119896119904+1=120587 (12 + 119873)

1minus119904

119904intinfin

0

(1 + 1199052)minus1199042

cos (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

120578 (119904 + 1) +119873

sum119896=1

(minus1)119896

119896119904+1=120587(minus1)119873(12 + 119873)

1minus119904

119904intinfin

0

sinh (120587119905 (12 + 119873)) (1 + 1199052)minus1199042

sin (119904 arctan (119905))cosh2 (120587119905 (12 + 119873))

119889119905

(72)



120577 (119904) minus119873

sum119896=1

1

119896119904= minus

1

2119873119904+ 1198731minus119904

intinfin

0

cosh (120587119873119905) (1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 120590 gt 1 (73)

= minus1

2119873119904+1198731minus119904

119904 minus 1+ 21198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))1198902119873120587119905 minus 1

119889119905 (74)

120578 (119904) +119873

sum119896=1

(minus1)119896

119896119904=(minus1)119873

2119873119904+ (minus1)

1+119873

1198731minus119904

intinfin

0

(1 + 1199052)minus1199042

sin (119904 arctan (119905))sinh (120587119873119905)

119889119905 (75)







120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120577 (119904) =sin (1205871199042) 1205872


times intinfin

0

1199052minus119904

1F2(minus

119904

2+ 1

3

2 2 minus

119904

211990521205872

4)


(76)


120577 (119904) = (120587

21minus119904 minus 1) sin(120587119904

2)infin

sum119896=0

1205872119896

Γ (2119896 + 2)

times intinfin

0

1199051minus119904+2119896

sinh2 (120587119905)119889119905 120590 lt 0

(77)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)


timesinfin

sum119896=0


Γ (2119896 + 2)

(78)


120577 (119904) =2119904120587119904minus1 sin (1205871199042)

21minus119904 minus 2 + 119904

timesinfin

sum119896=1


Γ (2k + 2)

(79)


120577 (119904) =120587119904minus1

1 minus 21minus119904sin(120587119904

2)

timesinfin

sum119896=1


Γ (2119896 + 2)

120590 gt 0

(80)


120577 (119904) =1

Γ (119904) (2119904 minus 1 minus 119904)

timesinfin

sum119896=1

2minus2119896120577 (119904 + 2119896) Γ (1 + 119904 + 2119896)

Γ (2119896 + 2)

(81)




sum119896=0

(minus12058724)119896

120578 (119904 minus 2119896)

Γ (119896 + 12) Γ (119896 + 1) 120590 gt 1 (82)


120577 (119904)

= minus1

2

suminfin

119896=1(minus1205872)

119896



(83)


120577 (2) = minus1205872

3120577 (0) (84)


120577 (119904) =1

2(120587119904

infin

sum119896=1


Γ (2119896 + 1) Γ (119904 minus 2119896)


minus 1) cos(1205871199042))minus1

) 120590 gt 1

(85)


52 By Splitting


120577minus(119904) = int

120596

0

119905119904

sinh2 (119905)119889119905 120590 gt 1 (86)

120577+(119904) = int

infin

120596

119905119904

sinh2 (119905)119889119905 (87)

with

120577 (119904)Γ (119904 + 1)

2119904minus1= 120577minus(119904) + 120577

+(119904) (88)




infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












infin

sum119896=0

11986121198961205962119896+119904minus1

Γ (119896 minus 12) Γ (119896 + 1) (2119896 minus 1 + 119904) (89)


120577minus(119904) = 4120596

119904minus1

infin

sum119896=0

(minus12059621205872)119896

Γ (119896 + 12) 120577 (2119896)

Γ (119896 minus 12) (2119896 minus 1 + 119904) 120596 le 120587

(90)


120577minus(119904) =

2119904120596119904+1

1205872 (119904 + 1)

timesinfin

sum119898=1

1

1198982 21198651(1

1

2+119904

23

2+119904

2 minus

1205962

12058721198982)

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(91)



120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

Γ (119896 + 1)

Γ (119896 + 32 + 1199042)

infin

sum119898=1

(1

120587211989821205962 + 1)119896+1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(92)


120581 (119903 119901 119905) equivinfin

sum119898=1

119898119903

(1 + 11990521198982)119901

=infin

sum119898=0


Γ (119901) Γ (119898 + 1)

(93)


infin

sum119898=0

1

(11989821199052 + 1199102)=

1

21199102+

120587

2119905119910coth(

120587119910

119905) (94)




120577minus(119904) = 119904120596

119904minus1

Γ (1

2+119904

2)

timesinfin

sum119896=0

infin

sum119898=0

(minus1)119898Γ (119898 + 119896 + 1) 120577 (2119898 + 2119896 + 2)

Γ (119896 + 32 + 1199042) Γ (119898 + 1)

times (1205962

1205872)

(119898+119896+1)

minus 120596119904coth (120596) + 119904120596119904minus1

119904 minus 1

(95)


120577minus(119904) =

2119904120596119904minus1

Γ (12 + 1199042)

infin

sum119896=0

(1205962

1205872)

(119896+1)

timesinfin

sum119898=0

(1205962

1205872)

119898

(minus1)119898

Γ (119898 + 119896 +1

2+119904

2)

times 120577 (2119898 + 2119896 + 2)

times (Γ (119898 + 1) Γ (119896 + 1) (2119896 + 1 + 119904))minus1

minus 120596119904 coth (120596) + 119904120596119904minus1

119904 minus 1

(96)


119896=0suminfin

119898=0119891(119898 119896) = sum

infin

119896=0sum119896

119898=0119891(119898 119896 minus


120577minus(119904) = minus2119904120596

119904minus1

infin

sum119896=0

(minus1205962

1205872)

119896

120577 (2119896)

(2119896 minus 1 + 119904)minus 120596119904 coth (120596)

(97)



119904(119911)


119864119904(119911) = 119911

119904minus1

Γ (1 minus 119904 119911) (98)


120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2120596119896) (99)



119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 Analytic 120596 = 1 120596 = 120587

0 minus1

2+120596 coth (120596)

20156518 1076674

1 minus1

2+1

4

1205962

sinh2120596+1

4120596 coth(120596) 0927424 times 10minus2 0306837

2 3120596

16coth3120596 minus

1

2coth2120596 +

120596 coth120596 (minus3 + 21205962)

16sinh2120596+

8 + 31205962

16sinh21205960795491 times 10minus3 0134296


119904(119911) to

obtain

119864minus119904(119911) =

1

1199112(eminus119911 (119904 + 119911) + 119904 (119904 minus 1) 119864

2minus119904(119911)) (100)


120577+(119904) = minus 119904120596

minus1+119904 log (1198902120596 minus 1) + 2119904120596119904

minus2120596119904

1 minus 1198902120596

+ 2120596119904

Γ (119904 + 1)119873+1

sum119896=2

Li119896(119890minus2120596)

Γ (119904 minus 119896 + 1) (2120596)119896

+2120596119904minus119873minus1Γ (119904 + 1)

Γ (119904 minus 119873 minus 1) 2119873+1

infin

sum119896=1

119864119873+2minus119904

(2119896120596)

119896119873+1

(101)






120577+(119904) = 4120596

119904+1

infin

sum119896=1

119896119864minus119904(2119896120596)

= 21minus119904

Γ (119904 + 1) 120577 (119904) minus4120596119904+1

119904 + 1

timesinfin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596)

(102)


120577minus(119904) =

4120596119904+1

119904 + 1

infin

sum119896=1

119896119890minus2119896120596

11198651(1 2 + 119904 2119896120596) (103)






+(119904) if (88) is used


minus(119904) and 120577

+(119904) anti-


21minus119904

Γ (119904 + 1) 120577 (119904)

= 4120596119904+1

(infin

sum119896=1

119896(119864minus119904(2119896120596) + 1

1198651(119904 + 1 2 + 119904 minus2119896120596)

119904 + 1))

(104)





infin

sum119896=1

1198641(2119896120596) = minus



minusinfin

sum119896=2


120587119896radicminus1205962

+1

2log(120596

120587) +

1

2120596+120574

2

(105)


sum119896=1

log(1 + 1




1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

2

infin

sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962=

1

2ln (2)2 minus

1205742

2+1205872

12


sum119896=1

1198641(2119896)

119896

(107)


infin

sum119896=1

1198641(2119896)

119896= minusint

infin

1


119905119889119905 (108)


infin

sum119896=1

119905119896

1198962120577 (2119896) = int

119905

0


V (109)


sum119896=1

120577 (2119896) (minus1)119896

12058721198961198962= minus2int

1

0

1

119905(log(1

2119890119905

minus1

2119890minus119905

) minus log (119905)) 119889119905

(110)


int1

0


2V)

119889V

V= minus

1205872

12+1205742

2+ 120574 (1)

minus1

2log2 (2) + 2

(111)



minus119904(119911) into


120577+(119904) = minus4120596

119904+1

infin

sum119896=1

1

2120587119894int119888+119894infin


Γ (minus119905) (2119896120596)119905

119904 + 1 + 119905119889119905 (112)


120577+(119904) = minus

4120596119904+1

2120587119894int119888+119894infin



119904 + 1 + 119905119889119905 (113)




Γ (119904 + 1) 120577 (119904)

2119904minus1120596119904+1= minus

1

2120587119894∮

Γ (119905 + 1) 120577 (119905)

(119904 minus 119905) 2119905minus1120596119905+1119889119905 (114)





120577 (119904) = minus1


0


(115)


120577 (119904) =1

2minus

119904

120587

times intinfin

0


(1+119904)2

119889119905

120590 gt 1

(116)


120577 (119904) =1

2+1

2

radic120587 sin (1205871199042) Γ (1199042 minus 12)

Γ (1199042)

+ 120587intinfin

0

119905 sin (119904 arctan (119905))sinh2 (120587119905)

times21198651(1

2119904

23

2 minus1199052

)119889119905


0


times21198651(1

2119904

23

2 minus1199052

)119889119905 120590 gt 0

(117)




21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











21198651(1

2119904

23

2 minus1199052

) = (1 + 1199052

)minus1199042

times21198651(1

119904

23

2

1199052

(1 + 1199052))

(118)




120577 (119904) =120587119904minus14119904 sin (1205871199042)2 minus 21+119904 + 1199042119904

times intinfin

0


119889119905

120590 lt 2

(119)


120577 (119904) =1205871199042119904minus2

(1 minus 2119904) cos (1205871199042)

times (sin (120587119904) 2119904

120587intinfin

0


119889119905

+1

Γ (119904)) 120590 lt 2

(120)

120577 (119904) = 120587119904minus1 sin(120587119904

2)

times (intinfin

0

Vminus119904minus1

(V


minusV119889V

minus120587

sin (120587119904) Γ (1 + 119904)) 120590 lt 2

(121)

8 The Case 119904 = 119899



120577 (2119899) = minus1

2(119899

sum119896=1


Γ (2119896 + 2))



)

(122)

from (83) we find

120577 (2119899) =1


times119899

sum119896=1

(minus1205872)119896


Γ (2119896 + 1)

(123)


120577 (2119899) = 22minus2119899


times119899

sum119896=1


Γ (2119896 + 2))

(124)


120577 (2119899) = minus(minus1205872)

119899

21minus2119899 + 2119899 minus 2

times (119899minus1

sum119896=0

(2119896 minus 1) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 2))

(125)

thought to be new

120577 (2119899) =(minus1205872)

119899


sum119896=0

(1 minus 21minus2119896) 120577 (2119896)

(minus1205872)119896

Γ (2119899 minus 2119896 + 1)) (126)


120577 (2119899) =2 (minus1205872)

119899


sum119896=0


(2120587) 2119896Γ (2119899 minus 2119896 + 2)) (127)





21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











21198651(1 119899 + 12 119899 + 32 minus120596212058721198982)

1198982

= (119899 +1

2)(

minus1205962

12058721198982)

minus119899

times (2120587

119898120596arctan( 120596

120587119898) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

12 + 119896)

(128)

and for the case 119904 = 2119899 + 3

21198651(1 119899 + 1 119899 + 2 minus120596212058721198982)

1198982

= (119899 + 1) (minus1205962

1205872

1198982

)minus119899

times (1205872

1205962ln(1 + 1205962

12058721198982) minus

1

1198982

119899minus1

sum119896=0

(minus120596212058721198982)119896

119896 + 1)

(129)


120577 (2) = minus 4120596infin

sum119898=1

(120587119898

120596arctan( 120596

120587119898) minus 1)

+ Li2(119890minus2120596

) minus 2120596 log (minus1 + 1198902120596

)

+ 31205962

+ 2120596

(130)



120577 (2) = 5 minus 4(infin

sum119898=1

(minus1)119898120587(minus2119898)120577 (2119898)

2119898 + 1)

+ Li2(119890minus2


(131)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894

4120587(6 + 4 log (2120587) + 2Li

2(119890minus2

)


3)

= 0232662175652953119894

(132)


log(119866 (1 + 119894120587)

119866 (1 minus 119894120587)) =

119894 log (2120587)120587

minus119894

120587

+ (infin

sum119896=1

(minus2119894

120587+ 119896 ln(119896 + 119894120587

119896 minus 119894120587)))

(133)




119891 (119904) = minus120587119904minus1 sin (1205871199042)

119904 minus 1 (134)

119892 (119904) =2119904minus1

Γ (119904 + 1) (135)

ℎ (119905) = radic119905 (1

sinh2119905minus

1

1199052) (136)


120577 (119904) =2119891 (119904) 119892 (119904)

119892 (119904) minus 119891 (119904)

times intinfin

0


+2119894119891 (119904) 119892 (119904)

119891 (119904) minus 119892 (119904)

times intinfin

0


(137)


and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and simultaneously

120577 (119904) =2119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


minus2119894119891 (119904) 119892 (119904)

119891 (119904) + 119892 (119904)

times intinfin

0


(138)

Let


0

ℎ (119905) cos (120588 ln (119905)) 119889119905

119878 (120588) = intinfin

0

ℎ (119905) sin (120588 ln (119905)) 119889119905

(139)


1003816100381610038161003816100381610038161003816119891 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

=1003816100381610038161003816100381610038161003816119892 (

1

2+ 119894120588)

1003816100381610038161003816100381610038161003816

2

(140)


120577 (1

2+ 119894120588) =

2119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) + 119892 (12 + 119894120588)119862 (120588)

=2119894119891 (12 + 119894120588) 119892 (12 + 119894120588)

119891 (12 + 119894120588) minus 119892 (12 + 119894120588)119878 (120588)

(141)


120577 (1

2+ 119894120588) = 120577

119877(1

2+ 119894120588) + 119894120577

119868(1

2+ 119894120588) (142)


120577119877(12 + 119894120588)

120577119868(12 + 119894120588)

=119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587))

119862119901(120588) cos (120588 ln (2120587)) + 119862


(143)

where

119862119901(120588) = Γ

119877(1

2+ 119894120588) cosh (

120587120588

2) + Γ119868(1

2+ 119894120588) sinh(

120587120588

2)

(144)

119862119898(120588) = Γ

119868(1

2+ 119894120588) cosh (

120587120588

2) minus Γ119877(1

2+ 119894120588) sinh(

120587120588

2)

(145)


119877(12 + 119894120588) or 120577



119877(12 + 119894120588

0) = 120577

119868(12 + 119894120588

0) = 0) of


119878 (1205880) = 0

119862 (1205880) = 0

(146)



120577119877(12 + 119894120588)

1015840

120577119868(12 + 119894120588)

1015840= minus

1205771015840(12 + 119894120588)119868

1205771015840(12 + 119894120588)119877

= (119862119898(120588) cos (120588 ln (2120587)) minus 119862

119901(120588) sin (120588 ln (2120587)))

times (119862119901(120588) cos (120588 ln (2120587))


minus1

(147)






120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120578119877(120590 + 120588119894)

=1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(148)

120578119868(120590 + 120588119894)

= minus1

2120590(1minus120590)

intinfin

minusinfin

(1 + 1199052)minus1205902


119889119905

(149)

where

Ω (119905) =1

2120588 log (1205902 (1 + 119905

2

)) + 120590 arctan (119905) (150)




10 Comments



Cl2(119911) =

infin

sum119896=1

sin(119911119896)1198962

(152)


120587int119905

0


119866 (1 + 119905)) (153)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

1

2120587Cl2(2120587119905)

0 lt 119905 lt 1

(154)


log(119866 (1 minus 119905)

119866 (1 + 119905)) = 119905 log( sin (120587119905)

120587) +

119894120587

21198612(119905)

+eminus1198941205872

2120587Li2(e2120587119894119905)

(155)



infin

sum119896=1

120577 (2119896) 119905(2119896)

119896 + 1=1

2minus120587

1199052int119905

0

V2cot (120587V) 119889V (156)


infin

sum119896=1

120577 (2119896) 1199052119896

119896 + 1=

1

3119894120587119905 +

1


2119894120587119905

)

+ 119894Li2(1198902119894120587119905)

120587119905

minusLi3(1198902119894120587119905) minus 120577 (3)

212058721199052

(157)



Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Appendix



] (119904) = 119862 sin (120587 (119904 minus 1199040)) sin (120587 (119904 + 119904

0))


0)) sin (120587 (119904 + 119904

lowast

0))

(A1)


0 1199040

= 1205900+ 1198941205880is


119862 =1

(cosh2 (1205871205880) minus cos2 (120587120590

0))2 (A2)


120577] (119904) equiv ] (119904) 120577 (119904) (A3)


0) and




Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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