New integral representations. . The polylogarithm function

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1 New itegral repreetatio of the polylogarithm fuctio Djurdje Cvijović Atomic Phyic Laboratory Viča Ititute of Nuclear Sciece P.O. Box 5 Belgrade Serbia. Abtract. Maximo ha recetly give a excellet ummary of the propertie of the Euler dilogarithm fuctio ad the frequetly ued geeraliatio of the dilogarithm the mot importat amog them beig the polylogarithm fuctio Li ( ). The polylogarithm fuctio appear i everal field of mathematic ad i may phyical problem. We by makig ue of elemetary argumet deduce everal ew itegral repreetatio of the polylogarithm Li ( ) for ay complex for which <. Two are valid for all complex wheever Re >. The other two ivolve the Beroulli polyomial ad are valid i the importat pecial cae where the parameter i a poitive iteger. Our earlier etablihed reult o the itegral repreetatio for the Riema eta fuctio ζ ( + ) N follow directly a corollarie of thee repreetatio. Keyword: Polylogarithm itegral repreetatio Riema eta fuctio Beroulli polyomial Mathematic Subject Claificatio. Primary M99; Secodary 33E.

2 . Itroductio Recetly Maximo [] ha give a excellet ummary of the defiig equatio ad propertie of the Euler dilogarithm fuctio ad the frequetly ued geeraliatio of the dilogarithm the mot importat amog them beig the polylogarithm fuctio iclude itegral repreetatio erie expaio liear ad quadratic traformatio fuctioal relatio ad umerical value for pecial argumet. Li ( ). Thee Motivated by thi paper we have begu a ytematic tudy of ew itegral repreetatio for Li ( ) ice it appear that oly half a doe of them ca be foud i the literature. Amog kow repreetatio the followig t Li ( ) dt ( Re > ( )) t Γ() e ad log t log (/ t) Li ( ) dt dt ( Re > ) Γ() t( t ) Γ() t are bet kow [ pp. 3-3]. Here by makig ue of fairly elemetary argumet we deduce everal ew itegral repreetatio of the polylogarithm fuctio for ay complex for which <. Two are valid for all complex wheever Re > (ee Theorem). The other two give i Corollary are valid i the importat pecial cae where the parameter i a poitive iteger. Our earlier publihed reult [3] o the itegral repreetatio for the Riema eta fuctio ζ ( + ) N directly follow a corollarie of thee repreetatio (ee Corollary ).

3 . The polylogarithm fuctio The polylogarithm fuctio (alo kow a Joquiére fuctio) Li ( ) i defied for ay complex ad a the aalytic cotiuatio of the Dirichlet erie k Li ( ) ( < ; C) () k k which i abolutely coverget for all ad all iide the uit dic i the complex - plae. Sometime Li ( ) i referred to a the polylogarithm of order ad argumet mot frequetly however it i imply called the polylogarithm. The fuctio Li ( ) for fixed ha o pole or eetial igularitie aywhere i the fiite complex - plae ad for fixed ha o pole or eetial igularitie aywhere i the fiite complex - plae. It ha oly oe eetial igular poit at. For fixed the fuctio ha two brach poit: ad. The pricipal brach of the polylogarithm i choe to be that for which Li ( ) i real for real (whe i real) ad i cotiuou except o the poitive real axi where a cut i made from to uch that the cut put the real axi o the lower half plae of. Thu for fixed the fuctio Li ( ) i a igle-valued fuctio i the -plae cut alog the iterval ( ) where it i cotiuou from below. I the importat cae where the parameter i a iteger Li ( ) will be deoted by Li ( ) (or Li ( ) whe egative). Li ( ) ad all Li ( ) are ratioal fuctio i. Li ( ) i the polylogarithm of order (i.e. the -th order polylogarithm). The pecial cae i the ordiary logarithm Li ( ) log( ) while 3

4 the cae 3 4 are claical polylogarithm kow repectively a dilogarithm trilogarithm quadrilogarithm etc. For more detail ad a exteive lit of referece i which polylogarithm appear i phyical ad mathematical problem we refer the reader to Maximo []. The polylogarithm Li ( ) of order 3 i thoroughly covered i Lewi tadard text [4] while may formulae ivolvig Li ( ) ca be foud i [ pp. 3-3] ad [5]. Berdt treatie [6] ca erve a a excellet itroductory text o the polylogarithm (ad umerou related fuctio) ad a a ecyclopaedic ource.. Statemet of the reult I preparatio for the tatemet of the reult we eed to make everal defiitio. The Riema eta fuctio ad the Hurwit eta fuctio ζ () ad ζ ( a) repectively defied by mea of the erie [7 p. 87 Eq ad 3..]: are ζ () Re > ) ( k k k (k+ ) k ( ) ( Re > ; ) k k () ad [ pp. 4-7] ζ ( a ) ( Re > a --...). (3) ( k + a) k Both are aalytic over the whole complex plae except at where they have a imple pole. 4

5 Next for ay real x ad ay complex with Re > we defie i( kx) S ( x) (4a) k k ad co( kx) C ( x). (4b) k k We alo ue the Beroulli polyomial of degree i x B ( x ) defied for each oegative iteger by mea of their geeratig fuctio [7 p. 84 Eq. 3..] tx te t B ( x) ( t < ) t π (5a) e! ad give explicitly i term of the Beroulli umber B B () by [5 p. 765] k k k B( x) B k x. (5b) Our reult are a follow. Theorem. Aume that ad are complex umber let Li ( ) be the polylogarithm fuctio ad let S ( x ) ad C ( x) be defied a i Eq. (4a) ad (4b) repectively. If Re > ad < the 5

6 πt π (6a) πt + i( ) Li( ) S ( t) dt co( ) Li( ) C( πt) dt co( ) (6b) πt + ( π ) co( t) C ( πt) dt. (6c) co( ) πt + Here ad throughout the text we et either or /. Remark. Oberve that our itegral repreetatio i the above Theorem eetially ivolve either the Hurwit eta fuctio or the geeralied Claue fuctio. Ideed betwee the fuctio S ( x ) ad C ( x) ad the Hurwit eta fuctio defied a i (3) there exit the relatiohip [8 p. 76 Etry 5.4..] S ( πt) ( π ) cc( π / ) ζ ( t ) ζ ( t) C ( πt) 4 ( ) ec( π/ ) Γ ( t ; Re > ; N) + where Γ() i the familiar gamma fuctio. I the cae whe thi relatiohip doe ot hold S ( x ) ad C ( x) are ued to defie the geeralied Claue fuctio [4 p. 8 Eq. 3] C ( x) if i odd Cl( x) ( N). S( x) if i eve What i mot importat however i that S ( ) x ad C ( ) x ( N) are expreible i term of the Beroulli polyomial ad thi make poible our corollarie. 6

7 Corollary. Aume that i a poitive iteger let Li ( ) be the polylogarithm fuctio ad let B ( ) x be the Beroulli polyomial of degree i x give by Eq. (5). The provided that < we have ( π) i( πt) ( ) ( ) ( ) ( )! co( πt) + Li B t dt (7a) ( π ) ( ) ( ) ( ) ( )! co( πt) + Li B t dt (7b) ( π ) co( t) ( π ) ( ) B ( t) ( )! co( πt) + dt. (7c) Corollary. Aume that i a poitive iteger let ζ () be the Riema eta fuctio defied a i () ad let B ( x) be the Beroulli polyomial of degree i x give by Eq. (5). We the have: ζ + ( π ) (+ ) ( ) B + ( )cot( ) ( )! + t π t dt (8a) ζ + ( π ) (+ ) ( ) B ( ) ta( ) + π ( )! + t t dt. (8b) Remark. The itegral for ζ ( + ) i (8a) whe i well kow [7 p. 87 Eq. 3..7] while the other itegral i Corollary were recetly deduced [3 Theorem ]. However we have failed to fid the itegral repreetatio give i Theorem ad Corollary i the literature. 7

8 Remark 3. The itegral repreetatio give i (6) ad (7) hold for <. However our reult may be exteded to ay complex > by mea of the iverio formula [ pp. 3-3; 5 pp ] ( π ) log( ) Li e e Li Γ() π i iπ/ iπ ( ) ζ + () where ζ ( a) i the Hurwit eta fuctio (3) or by mea of the followig particularly imple iverio formula ( π i) log Li( ) ( ) Li B! π i valid for the -th order polylogarithm. Remark 4. We remark that the exitece of the itegral i (8) i aured ice the itegrad o [ α] < α < have oly removable igularitie. Thi ca be demotrated eaily by makig ue of ome baic propertie of B ( x ). For itace kowig that the odd-idexed Beroulli umber B apart from B / are ero [7 p. 85 Eq. 3..9] we have B () t (+ ) B () t lim B ( t) ta( πt) lim lim ( )(+ ) B / π co( πt) π i( πt) + + t / t / t / ice B (/ ) ( ) B [7 p. 85 Eq. 3..] 3. Proof of the reult I what follow we hall eed the followig lemma ad we provide two proof for it. The ecod proof wa uggeted by oe of the aoymou referee ad it make ue of wellkow Chebyhev polyomial ad Fourier erie. 8

9 Lemma. Aume that i a poitive iteger ad that ad ½. The we have: co( πt) i( πt) dt i( πt) co( πt) + (9a) co( πt) dt i( πt) co( πt) +. (9b) Firt Proof of Lemma. It i clear that the itegral i (9) ca be rewritte a follow: co( t) it π dt i( t) cot + (a) π co( t) π dt i( t) cot + π. (b) We firt etablih the cae i (). For ay poitive iteger ad arbitrary complex coider the itegral I ad I with parameter give by ( ) I π it it e dt cot+ π it co( t) dt + co t+ π it i( t) dt (a) co t+ i ad by I ( ) π it e dt cot+ 9

10 π co( t) dt + co t+ π i( t) dt. (b) co t+ i I order to derive the itegral I ad I we make ue of cotour itegratio ad calculu of reidue. By ettig cot τ / τ ) it τ e ( i τ + ad i t ( / τ ) where i we arrive at ( ) I τ iτ ( τ ) dτ ( τ )( τ / ) iτ τ τ ( τ ) dτ ( τ )( τ / ) τ ( τ ) πi Re τ ( τ )( τ / ) π i (a) ad ( ) I τ + τ ( ) dτ ( τ )( τ / ) iτ τ τ ( ) dτ i( τ )( τ / ) τ ( ) πi Re τ i( τ )( τ / ) ( πi i ) π. (b) Combiig () ad () together ad equatig the real ad imagiary part o both ide give the itegral i () where. I thi way we evaluate the itegral i (9) where. Oberve that i both itegral i () the cotour i the uit circle ad i travered i the poitive (couterclockwie) directio ad the oly igularitie of the itegrad that lie iide the cotour are at τ.

11 Next it ca be eaily how that the cae / i () reduce to the above coidered cae. Ideed i view of the followig well-kow property [8 p. 7 ad 73 Eq.... ad...] a f () tdt f( a t) f( t) a f( t) dt f( a t) f( t) it uffice to how that both itegrad i () are uch that f ( π t) f( t). Thi complete the proof of our lemma. Secod Proof of Lemma. Coider the Chebyhev polyomial of the firt ad ecod kid T ( m x ) ad Um( x) defied by [7 p. 776 Eq..3.5 ad.3.6] T (co θ ) co( mθ ) ad U (co θ ) i[( m+ ) θ]/ iθ m m ad recall that their geeratig fuctio are give by [7 p. 783 Eq..9.9 ad.9.] m + T ( ) m x x + m x m Um( x) ( < ). + m Now by applyig the orthogoality relatio of the trigoometric fuctio we for itace have the ought formula i (9b) π co( πt) co( t) dt i( πt) co( πt) + π i( t) cot+ dt π co( t) π m co( t) Tm (co t) dt π i( t) + m π i( t) dt π m co( t) + co( mt) dt m π i( t). The itegral give i (9a) follow i a imilar maer.

12 Proof of Theorem. The proof of Theorem ret o the above Lemma. From (9a) we obtai the formula k i( πt) i( k πt) dt /; k.... co( πt) + Dividig the formula by k ad ummig over k we have k i( k πt) i( πt) Re dt k k k > k co( πt) + o that k i( k πt) i( πt) dt Re k k >. (3) k k co( πt) + Now the required itegral formula i (6a) directly follow from the lat expreio i light of the defiitio of Li ( ) i () ad S ( x) i (4a). It hould be oted that ivertig the order of ummatio ad itegratio o the right-ide of (3) i jutified by abolute covergece of the erie ivolved. Startig from (9b) the formula i (6b) i derived i preciely the ame way. I order to prove (6c) ote that ( π ) co( t) + co( πt) + co( πt) + thu we have

13 ( π ) co( t) C ( πt) dt C ( πt) d t co( πt) + co( πt) + ice C ( πt) dt co k t dt ( π ). k k Thi prove the theorem. Proof of Corollary. We eed the Fourier expaio for the Beroulli polyomial B ( x ) [7 p. 85 Eq. 3..7] B ( )! i( kπ x) ( x) ( ) ( π ) k k where x for 3 < x < for ad [7 p. 85 Eq. 3..8] B ( )! co( kπ x) ( x) ( ) ( π ) k k where x for. Thee Fourier expaio ca be rewritte a follow ( π ) S ( x) ( ) B ( x) ( )! ( π ) C( x) ( ) B( x) ( )! i term of the fuctio S ( x ) ad C ( x) defied i (4a) ad (4b). Fially the itegral formulae for -th polylogarithm propoed i (7) are obtaied from the above expaio i cojuctio with Theorem. 3

14 Proof of Corollary. Firt ote that for ad - the polylogarithm reduce to the Riema eta fuctio Li () ζ ( ) ( Re > ) (4a) Li ( ) ( ) ζ ( ) ( ). (4b) Secodly it i ot difficult to how that i( πt) lim cot( πt) co( πt) + (5a) ad i( πt) lim ta( πt). (5b) co( πt) + Fially we deduce the itegral formulae i (8) tartig from (7a) ad by employig the expreio i (4) ad i (5). 4

15 Referece [] L. C. Maximo The dilogarithm fuctio for complex argumet Proc. R. Soc. Lod. A 459 (3) [] A. Erdélyi W. Magu F. Oberhettiger ad F. G. Tricomi Higher tracedetal fuctio volume I McGraw-Hill New York 953. [3] D. Cvijović ad J. Kliowki Itegral repreetatio of the Riema eta fuctio for odd-iteger argumet J. Comp. App. Math. 4 () [4] L. Lewi Polylogarithm ad Aociated Fuctio North-Hollad New York 98. [5] A. P. Prudikov Yu. A. Brychkov ad O. I. Marichev Itegral ad Serie volume III: More Special Fuctio Gordo ad Breach New York 99. [6] B. C. Berdt Ramauja' Notebook Part I Spriger-Verlag New York 985. [7] M. Abramowit ad I. A. Stegu Hadbook of Mathematical Fuctio with Formula Graph ad Mathematical Table Dover New York 97. [8] A. P. Prudikov Yu. A. Brychkov ad O.I. Marichev Itegral ad erie volume I: Elemetary Fuctio Gordo ad Breach New York