Euler s Integrals and Multiple Sine Functions

Transcription

1 Euler s Itegrals ad Multiple Sie Fuctios Shi-ya Koyama obushige Kurokawa Ruig title Euler s Itegrals Abstract We show that Euler s famous itegrals whose itegrads cotai the logarithm of the sie fuctio are expressed via multiple sie fuctios Mathematics Subject Classificatio: M6 Itroductio Euler studied the defiite itegrals x logsi xdx for = ad I [E] 769, he proved the famous result logsi xdx = π log, which is frequetly explaied as a example of tricky itegrals i aalysis courses A bit later, Euler [E] 77 stated that x logsi xdx = π log Euler proved by usig logsi x = :odd = 7 π ζ log 6 8 cosx for < x < π The actual itegratio is obvious sice cosxdx = log

2 for =,,, Hece, the origial proof of Euler is ot tricky cotrary to the usual explaatio I the case of, Euler primary wated to calculate the value ζ He started from the diverget series expressio ζ = π ζ = π log ad by ivestigatig it he reached Thus his argumets are difficult to follow ad literally ivalid Moreover Euler did ot prove the fuctioal equatio ζ = π ζ cojectured by himself i [E] We otice that whe we use we ca give a secure calculatio for : with itegratio by parts x logsi xdx = x cosxdx = x cosxdx π 8 log It might be remarkable that Euler missed this way I this paper we ivestigate defiite itegrals x θ r logsi θdθ if is odd, if is eve for r =,,, cotaiig the Euler s case x = π/ from the poit of view of multiple sie fuctios Let S r x = e xr r = e xr r = P r x P r x r P r x r r be the multiple sie fuctio studied i [K, K, KK, KK, KOW, KW], where P r u = u exp u + u + + ur r

3 For example ad S x = e x S x = e x S x = e x The we show the followig result x + x e x, x e x x + x e x+ x Theorem For x < π ad for r =,,,, we have x I particular we have: Theorem For r =,,,, Examples θ r logsi θdθ = xr πr x logsi x r r log S r π θ r logsi θdθ = πr r log S r logsi θdθ = π log S = π log e θ logsi θdθ = π log S = π log e 8 θ logsi θdθ = π log S = π log e e, + e, 5 exp + + 6

4 We otice that we have S = ad S = exp 7ζ from Euler s results 8π ad combied with ad 5 We demostrate a calculatio of the special value S from the product expressio directly as follows Theorem ad S = 8 exp 9ζ 6π θ logsi θdθ = π π ζ log 6 I our calculatio a geeralizatio of the Stirlig formula is crucial Ackowledgemet We thak the referees for their suggestios to refie the presetatio Multiple Sie Fuctios To make this paper self-cotaied we prove some basic properties of multiple sie fuctios For geeral backgroud we refer to [KK, KK, KOW, M] Propositio For r =,,,, S r x is a meromorphic fuctio i x C ad it satisfies S rx S r x = πxr cotπx Proof The fact that S r x is meromorphic fuctio i x C ad its order as a meromorphic fuctio beig r is see from the product expressio defiig S r x Let us calculate the logarithmic derivative From we have S r x = e xr r log S r x = xr r + = xr r + x + + x x + r x P r P r x r r x r log P r + r log P r x log r x + r log + x x r r x + + r r x

5 Hece S rx S r x = xr + r x + r x x + + xr r + r + x xr + + r r Here ad Thus where we used + x + + xr = x r r x + x xr + + r = r x r r x + S rx S r x = x r + = x r + = πx r cotπx, x r r x x r x + x r x π cotπx = x + x x Propositio For x < ad for r =,,,, log S r x = x πt r cotπtdt Proof Sice S r =, both sides are at x = Hece it is sufficiet to remark that the differetiatios of both sides are πx r cotπx from Propositio Euler s Itegrals Usig Propositio we show Theorems ad Proof of Theorems ad : By itegratio by parts i log S r x = x πt r cotπtdt 5

6 we have log S r x = [ t r logsi πt ] x x r t r logsi πtdt = x r logsi πx r x Hece chagig the variable to θ = πt i the itegral, we have log S r x = x r logsi πx r π r x This gives Theorem The, lettig x = / we have Theorem t r logsi πtdt θ r logsi θdθ Examples: logsi θdθ = π 8 log π log S logsi θdθ = π log π log S log S θ logsi θdθ = π π log 6 θ logsi θdθ = π 8 log π log S,,, A calculatio of the special value Proof of Theorem : Sice S = e exp +, + we put A = e = exp exp

7 ad show lim A = 9 8 exp ζ = 8 exp 9ζ 6π The the value of the itegral follows from 6 We use the Stirlig formula ad its geeralizatio! = π + e exp ζ e 9 +, which follow from the Euler-Maclauri summatio formula for ζ s: ζ s = lim s log + s log s s s + s log s s log + s valid i Res > We refer to Hardy [H] Chap XIII for the Euler-Maclauri summatio formula ad its applicatios The, lettig s = ad we see log π = ζ = lim log + log = lim log! + e ad ζ = lim log = lim log e 9 + log 9 + 7

8 Usig = = = + ad the geeralized Stirlig formulas e ζ e , = e ζ e 9 +, we have = π + e, π + e, A exp e 9 ζ e e + 8

9 Hece, combiig with + = + = exp exp log + = exp + 8, + we obtai the desired result lim A = 9 8 exp ζ Remarks The fact S = is also proved as Theorem ad we get agai from I fact: S = e e + = lim e e + = lim e 5 e = lim e 5 e + e = lim = by the usual Stirlig formula!! + e 9

10 The case of S / is similar by usig the geeralized Stirlig s formula: S = e 8 e = lim e 8 e = lim e + 8 = lim e / = 7 exp ζ Thus we obtai Euler s formula via 5! +! + + Except for S / = we do ot kow the algebraicity of S r / for r I fact we caot dey eve the optimistic expectatio S r / Q Refereces [E] L Euler De summis serierum umeros Beroulliaos ivolvetium ovi commetarii academiae scietiarum Petropolitaae [Opera Omia I-5, pp9- ] [E] L Euler Exercitatioes aalyticae ovi commetarii academiae scietiarum Petropolitaae [Opera Omia I-5, pp-67] [E] L Euler Remarques sur u beau rapport etre les series des puissaces tat directes que reciproques Mémoires de l académie des scieces de Berli Lu e 79 [Opera Omia I-5, pp 7-9] [H] GH Hardy Diverget Series Oxford Uiv Press 99 [K] Kurokawa Multiple sie fuctios ad Selberg zeta fuctios Proc Japa Acad 67A [K] Kurokawa Multiple zeta fuctios: a example Adv Stud i Pure Math

11 [KK] Kurokawa ad S Koyama Multiple sie fuctios Forum Math [KK] S Koyama ad Kurokawa Kummer s formula for multiple gamma fuctios J Ramauja MathSoc [KOW] Kurokawa, H Ochiai ad M Wakayama Multiple trigoometry ad zeta fuctios J Ramauja Math Soc 7 - [KW] Kurokawa ad M Wakayama O ζ J Ramauja Math Soc 6 5- [M] Yu I Mai Lectures o zeta fuctios ad motives accordig to Deiger ad Kurokawa Asterisque Hayabuchi, Tsuzuki-ku, Yokohama -5, Japa koyama@tmtvejp Departmet of Mathematics, Tokyo Istitute of Techology, 5-855, Japa kurokawa@mathtitechacjp