Chapter 8. Euler s Gamma function
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1 Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the Gamma fuctio. For t R >, z C, defie t z : e z log t, where log t is the ordiary real logarithm. Euler s Gamma fuctio is defied by the itegral Γ(z : e t t z dt (z C, Re z >. Lemma 8.. Γ(z defies a aalytic fuctio o {z C : Re z > }. Proof. This is stadard usig Theorem Let U : {z C : Re z > }. First, the fuctio F (t, z : e t t z is cotiuous, hece measurable o R > U. Secod, for each fixed t >, z e t t z is aalytic o U. Third, let K be a compact subset of U. The there exist δ, R > such that δ Re z R for z K. This implies that for z K, t >, { t e t t z δ for t, M(t : e t t R Ce t/2 for t >, where C is some costat. Now we have M(tdt t δ dt + C e t/2 dt δ + 2C <. Hece all coditios of Theorem 2.23 are satisfied, ad thus, Γ(z is aalytic o U. 2
2 Usig itegratio by parts, oe easily shows that for z C with Re z >, that is, Γ(z z e t dt z ( z [e t t z t t + (8. Γ(z + zγ(z if Re z >. e t t z dt z Γ(z +, Oe easily shows that Γ( ad the by iductio, Γ( (! for Z >. We ow show that Γ has a meromorphic cotiuatio to C. Theorem 8.2. There exists a uique meromorphic fuctio Γ o C with the followig properties: (i Γ(z e t t z dt for z C, Re z > ; (ii the fuctio Γ is aalytic o C \ {,, 2,...}; (iii Γ has a simple pole with residue ( /! at z for,, 2,...; (iv Γ(z + zγ(z for z C \ {,, 2,...}. Proof. The fuctio Γ has already bee defied for Re z > by e t t z dt. By Corollary 2.2, Γ has at most oe aalytic cotiuatio to ay larger coected ope set, hece there is at most oe fuctio Γ with properties (i (iv. We proceed to costruct such a fuctio. Let z C with Re z >. By repeatedly applyig (8. we get (8.2 Γ(z Γ(z + for Re z >,, 2,.... z(z + (z + We cotiue Γ to B : C \ {,, 2,...} as follows. For z B, choose Z > such that Re z + > ad defie Γ(z by the right-had side of (8.2. This does ot deped o the choice of. For if m, are ay two itegers with m > > Re z, the by (8.2 with z +, m istead of z, we have Γ(z + Γ(z + m, z + (z + m ad so z(z + (z + Γ(z + Γ(z + m. z(z + (z + m 22
3 Hece Γ is well-defied o B, ad it is aalytic o B sice the right-had side of (8.2 is aalytic if Re z + >. This proves (ii. We prove (iii. By (8.2 we have lim (z + Γ(z z lim Γ(z + + z z(z + (z + ( Γ(. ( ( + (! Hece Γ has a simple pole at z of residue ( /!. We prove (iv. Both fuctios Γ(z + ad zγ(z are aalytic o B, ad by (8., they are equal o the set {z C : Re z > } which has limit poits i B. So by Corollary 2.2, Γ(z + zγ(z for z B. Theorem 8.3. We have Γ(zΓ( z π si πz for z C \ Z. Proof. We prove that zγ(zγ( z πz/ si πz, or equivaletly, (8.3 Γ( + zγ( z πz si πz for z A : (C \ Z {}, which implies Theorem 8.3. Notice that by Theorem 8.2 the left-had side is aalytic o A, while by lim z πz/ si πz the right-had side is also aalytic o A. By Corollary 2.2, it suffices to prove that (8.3 holds for every z i a ifiite subset of A havig a limit poit i A. For this ifiite set we take S : { : Z 2 >}; this set has limit poit i A. Thus, (8.3, ad hece Theorem 8.3, follows oce we have proved that (8.4 Γ( + 2 Γ( 2 π/2 si π/2 (, 2,.... Notice that Γ( e s s /2 ds e t t /2 dt e s t (s/t /2 dsdt.
4 Defie ew variables u s + t, v s/t. The s uv/(v +, t u/(v +. The Jacobia of the substitutio (s, t (u, v is It follows that (s, t (u, v s u t u Γ( + 2 Γ( 2 s v t v uv u (v + 3 u (v + 2. e u v /2 u (v + dudv 2 v u v + (v + 2 v + u (v + 2 e u v /2 (s, t (u, v dudv e u udu v /2 (v + 2 dv. I the last product, the first itegral is equal to, while for the secod itegral we have, by homework exercise 4, v /2 (v + dv 2 [ ] v /2 + v + This implies (8.4, hece Theorem 8.3. Corollary 8.4. Γ( 2 π. ( v /2 d v + v + dv/2 Proof. Substitute z 2 i Theorem 8.3, ad use Γ( 2 >. dw w 2 + π/2 si π/2. Corollary 8.5. (i Γ(z for z C \ {,, 2,...}. (ii /Γ is aalytic o C, ad /Γ has simple zeros at z,, 2,.... Proof. (i Recall that Γ( (! for, 2,.... Further, by Theorem 8.3 we have Γ(zΓ( z si πz π for z C \ Z. (ii By (i, the fuctio /Γ is aalytic o C \ {,, 2,...}. Further, at z,, 2,..., Γ has a simple pole, hece /Γ is aalytic ad has a simple zero. 24
5 We give aother expressio for the Gamma fuctio. Theorem 8.6. For z C \ {,, 2,...} we have! z Γ(z lim z(z + (z +. Proof. Deote the limit by F (z. We prove by iductio that for every o-egative iteger we have F (z Γ(z for z C with Re z > ad (if > z,,...,. For the momet, we assume that this assertio is true for, i.e., F (z Γ(z for Re z >, ad do the iductio step. Assume our assertio holds for some iteger. Let z C with Re z > ad z,...,. We do ot kow a priori whether the limit F (z exists, but at least we have F (z + zf (z! z+ lim (z + (z + + ( z+ lim. + (z + (z + + ( +!( + z By the iductio hypothesis we kow that F (z + Γ(z +, ad so F (z does exist, ad F (z + Γ(z + F (z Γ(z. z z This completes the iductio step. We ow show that F (z Γ(z for z C, Re z >. For this, we eed some lemmas. Lemma 8.7. Let z C with Re z >. The! z z(z + (z + Proof. By substitutig s t/, the itegral becomes ( t t z dt. The rest is left as a exercise. z ( s s z ds. Lemma 8.8. For every iteger 2 ad every real t with t we have e t ( t 25 e t t2 2.
6 Proof. This is equivalet to t2 et( t ( t, 2. Recall that if f, g are cotiuously differetiable, real fuctios with f( g( ad f (x g (x for x A, say, the f(x g(x for x A. From this observatio, oe easily deduces that + x e x, x e x, ( x r rx for x, r. This implies o the oe had, for 2, t, o the other had e t( t e t( t e t (e t/, ( t ( t ( t 2 t Completio of the proof of Theorem 8.6. Let z C with Re z >. We prove that F (z Γ(z. By the itegral expressio for Γ(z ad by Lemma 8.7 we have { } ( Γ(z F (z lim e t t z t t dt z dt lim ( e t ( t t z dt. Now usig ad Lemma 8.8 we obtai Γ(z F (z lim lim e t t2 tz dt e t t Re z+ dt lim Γ(Re z + 2. We deduce some cosequeces. Recall that the Euler-Mascheroi costat γ is give by ( N γ lim log N. N 26
7 Corollary 8.9. We have Γ(z e γz z e z/ + z/ for z C \ {,, 2,...}. Proof. Let z C \ {,, 2,...}. The for N Z > we have Γ(z lim N N z N! z(z + (z + N z z lim N e(log N 2 N z e γz z e z/ + z/. N lim N e z/ + z/ e z log N ( + z( + z/2 ( + z/n As aother cosequece, we derive a ifiite product expasio for si πz. Corollary 8.. We have si πz πz ( z2 2 for z C. Proof. For z C we have by Theorem 8.3, Corollary 8.5 ad Corollary 8.9, si πz π Γ(zΓ( z π( z e γz z πz π Γ(z( zγ( z ( (e z/ + z ( z ( + z πz ( ze γz ( z2 2. ( (e z/ z Recall that the Beroulli umbers B ( are give by z e z B! z ( z < 2π. 27
8 Corollary 8.. We have B, B 2, B 3 B 5 ad ζ(2 ( 2 2 B 2 (2! π2 for, 2,.... Proof. Let z C with < z <. The si πz ad so, by takig the logarithmic derivative of si πz, (8.5 si πz si πz π cos πz si πz π(eπiz + e πiz /2 (e πiz e πiz /2i πi + z 2πiz e 2πiz πi + B z! (2πi z. We obtai aother expressio for the logarithmic derivative of si πz by applyig Corollary 2.28 to the product idetity from Corollary 8.. Note that for z C with z < we have z 2 / 2 < 2 ad that 2 coverges. Hece the logarithmic derivative of the ifiite product is the ifiite sum of the logarithmic derivatives of the factors, i.e., (8.6 si πz si πz (πz πz z + + z 2 ( z 2 / 2 z 2 / 2 2z/ 2 z 2 / 2 z 2 z 2 k z 2k+ 2k+2 z 2 ζ(2k + 2z 2k+. k ( k ( z 2 k 2 (by absolute covergece Now Corollary 8. easily follows by comparig the coefficiets of the Lauret series i (8.5 ad (8.6. We fiish with aother importat cosequece of Theorem 8.6, the so-called duplicatio formula. 28
9 Corollary 8.2. We have Γ(2z 22z π Γ(zΓ(z + 2 for z C, z, 2,, 3 2, 2,.... Proof. Let A be the set of z idicated i the lemma. We show that the fuctio F (z : 2 2z Γ(zΓ(z + /Γ(2z is costat o A. Substitutig z gives that the 2 2 costat is 2 π, ad the Corollary 8.2 follows. Let z A. To get ice cacellatios i the umerator ad deomiator, we use the expressios Γ(z lim! z z(z + (z + lim Γ(z + lim! z+/2 2 (z + /2(z + 3/2 (z + + /2 Thus, (2 +! (2 + 2z Γ(2z lim 2z(2z + (2z F (z 22z Γ(zΓ(z + 2 Γ(2z { 2 2z (! 2 2z+/2 lim 2z(2z + (2z { (! 2 } lim (2 +! sice lim This shows that ideed F (z is costat. 2 +! z 2z(2z + 2 (2z + 2, 2 +! z+/2 lim (2z + (2z + 3 (2z + 2 +, 2 2z 2z lim (2 + 2z e2z log(2/(2+. (limit over the odd itegers. } 2z(2z + (2z (2 +! (2 + 2z Remark. More geerally, oe ca derive the multiplicatio formula of Legedre- Gauss, (2π ( /2 Γ(z z /2 Γ(zΓ(z + Γ(z + 29
10 for every iteger 2. The idea of the proof is similar to that of Corollary 8.2 (exercise. 3
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