THE GAMMA FUNCTION. z w dz.

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1 THE GAMMA FUNCTION. Some results from lysis Lemm. Suppose f is sequece of fuctios lytic o ope subset D of C. If f coverges uiformly o every compct closed d bouded subset of D to the limit fuctio f the f is lytic o D. Moreover, the sequece of derivtives f coverges uiformly o compct subsets of D to f. Proof. Sice f is lytic o D we hve by Cuchy s itegrl formul f w = f z πi z w dz where is y closed d positively orieted cotour i D d w is y iterior poit. The regio iterior to d icludig is closed d bouded d hece compct. So f coverges uiformly o this regio d hece we c pss to the limit uder the itegrl sig givig fw = πi fz z w dz. This implies tht f is lytic o the regio defied by d hece o the whole of D. For the derivtives we hve d Hece f w = πi f w = πi f w f w = π f z z w dz fz z w dz. f z fz dz z w legth of sup z f z fz z w d this teds to s for y w o the iterior of, d hece for y compct subsets of D just choose ppropritely which is possible sice D is ope.

2 THE GAMMA FUNCTION Lemm Differetitig uder the itegrl sig. Let D be ope set d let be cotour of fiite legth L. Suppose ϕ : {} D C is cotiuous fuctio d defie g : D C by gz = ϕw, zdw. The g is cotiuous. Also, if ϕ z lytic with derivtive g z = exists d is cotious o {} D the g is ϕ w, zdw. z Proof. Let z, z D with z fixed. Sice ϕ is cotiuous, give ε > we c fid δ > such tht z z < δ ϕw, z ϕw, z < ε/l. Hece, give ε > choose δ s bove the by lierity of the itegrl d the estimtio lemm gz gz = ϕw, z ϕw, z dw L mx ϕw, z ϕw, z w < ε. Hece g is cotiuous. If ϕ exists d is cotiuous the z ϕw, z + h ϕw, z ϕ w, z h z with h. The gi by lierity of the itegrl d the estimtio lemm gz + h gz ϕ w, zdw h z = ϕw, z + h ϕw, z ϕ w, z dw h z L mx ϕw, z + h ϕw, z w ϕ w, z h z d this teds to with h. Corollry. Let D be ope set d ϕ : [, ] D C be cotiuous with cotiuous prtil derivtive ϕ. If the itegrl ϕt, zdt coverges uiformly o z compct subsets of D the it defies lytic fuctio there d hs derivtive t, zdt. ϕ z Proof. Let f z = ϕt, zdt so is the stright lie joiig d. By the bove lemm ech f is lytic with f z = ϕ t, zdt d by hypothesis z f f = ϕt, zdt uiformly o compct subsets of D. Applyig Lemm gives us the result.

3 THE GAMMA FUNCTION 3. The Alytic Chrcter of Γs Let s = σ + it with σ, t R. We defie the Γ-fuctio for σ > by Γs = e t t s dt. Note i the defiitio we hve the two bd poits, d. Also, we cot immeditely pply Corollry sice the itegrd is ot lwys cotiuous t. It turs out oe of this mtters d we hve the followig. Propositio. Γs is lytic for σ >. Proof. First, ote for > the fuctio defied by e t t s dt is lytic. To see this we oly eed show it is uiformly coverget o compct blh d the we c pply Corollry sice ll other hypotheses re met. As expected, the expoetil domites the til of the itegrl givig e t t s dt e t t s dt = e t t s dt C = Ce e t t σ dt e t dt d this s givig uiform covergece. Now for σ > defie, f s = e t t s dt. By the bove rgumet ech f is lytic. Suppose σ c >. For < t we hve e t < d t σ t c. Hece, for > m, m e t t s dt < m t c d = c m. Give ε > we c choose < δ < such tht c m < ε wheever m < δ. Hece the f stisfy the Cuchy coditio for uiform covergece i compct subsets of the hlfple σ >. Applyig Lemm we see tht the gmm fuctio is lytic for σ >. We c show the Γ fuctio is extesio of fctoril fuctio to complex rgumets, vi the followig fuctiol equtio Propositio. For σ > we hve Γs + = sγs.

4 4 THE GAMMA FUNCTION Proof. Itegrtio by prts gives e t t s dt = e t t s + s e t t s dt = sγs By direct computtio we see Γ = d hece by iductio Γ + =! for ll positive itegers. The fuctiol equtio lso gives us the lytic cotiutio of Γ. Theorem. The Γ fuctio c be exteded over the whole complex ple to meromorphic fuctio with simple poles t the egtive itegers d zero. The residues of these poles re give by 3 Res s= Γs = Proof. By we hve 4 Γs =! Γs+ ss+s+ s+ for y positive iteger. Now Γs+ is lytic for σ > so the fuctio o the right is meromorphic for σ > with simple poles t,,,...,. Sice is rbitrry we re doe. By costructio this extesio of Γ stisfies. To clculte the residues we rewrite 4 s d proceed directly viz: Γs = Γs+ + ss+s+ s+ ResΓ; = lim s s + Γs+ + ss+s+ s+ Γs+ + = lim s ss+s+ s+ =! where we hve used Γ = i the umertor. From ow o whe we refer to the Γ-fuctio we me the meromorphic cotiutio. Heuristiclly we c thik of this s the limit i of the right hd side of 4, d this is i fct ot too fr from the truth.

5 THE GAMMA FUNCTION 5 3. Product Represettios of Γs Sice e x = lim + x it is ot uresoble to expect 5 Γs = lim t t s dt. We first prove this d the use it to give ltertive represettio of Γs, which c be thought of s the limit i of 4. Lemm 3. Formul 5 holds for σ >. Proof. Deote The Γs f s = f s = e t t t s dt. t t s dt + e t t s dt. The secod itegrl is just the til of the Γ-fuctio which s. We wt to show the first itegrl lso. This seems fesible sice the first fctor of the itegrd gets rbitrrily smll for icresig. Now, for y we hve + y e y y. For lrge set y = t/ the Hece e t t e t + t. t = e t e t t e t + t = e t t t. Now, if the whe <. Lettig = t / the for lrge t t. Therefore e t t t e t. This gives Γs f s e t t σ+ dt < Γσ +

6 6 THE GAMMA FUNCTION s sice Γσ + is fiite. We deduce the ltertive represettio of Γs s follows. Substitutig u = t/ we hve t t s dt = s u u s du = s s us u + u u s du s = s u u s du s = Tkig the limit s gives Propositio 3. For s,,... = s... ss +... s +! = ss + s + s. 6 Γs = lim! ss + s + s. u s + du This coverges for ll other s so gives us other meromorphic cotiutio of Γ. This formul is quite useful d hs few cosequeces. The first of which is Corollry Weierstrss Product. For s,,... we hve 7 Γs = e s + s e z/k s k k= where is the Euler-Mscheroi costt. Proof. For s,,... we hve Γs = lim! ss + s + s log = lim es s + s + s/ + s/ e slog = lim = e s s lim s k= + s k e z/k. e s s + s/ + s/

7 THE GAMMA FUNCTION 7 This formul clerly demostrtes the poles s well s givig us the fct tht Γs hs o zeros. Also, tkig the logrithm of the product, differetitig d the evlutig t s = gives Γ =. 4. The Reflectio d Duplictio Formule We c use 6 to prove the fmous reflectio d duplictio formule. For the first of these we eed lemm. Lemm 4. We hve the followig expsios 8 π cotπs = s + s d 9 Proof. Let siπs πs = = s = F s = s + s = s. s. with s C\Z. If s is er iteger we expect to see some firly lrge terms i the series but these will die out s icreses. This is eough to gurtee bsolute covergece: for > s we hve s s > / d hece > s s < > s which coverges. Addig i the fiite umber of other terms gives tht the series i 8 coverges bsolutely. Note this lso implies the series coverges uiformly o compct subsets d hece defies lytic fuctio o C\Z by Lemm. Splittig the summd ito prtil frctios we see Fs is periodic i σ with period. The pole t is simple d hs residue. By periodicity every poles is simple with residue. Therefore, the fuctio defied by fs = π cotπs F s is etire d periodic i σ with period. We show fs is bouded the pply Liouville s theorem. By periodicity it suffices to show f is bouded whe σ < d sice f is etire we eed to show it s bouded s t = Is ±. Now, π cotπs = πi eπis + e πis = πi + πi e πis e πis e πis. Sice e πis = e πt we hve lim t ± π cotπs = πi. For F s ote tht i the regio σ < we hve t s < t +. We lso hve s =

8 8 THE GAMMA FUNCTION σ t + iσt σ t = σ t + t + σ > t +. Hece F s t + t + = + t dx + t + t x + t = x/ t + t + t t t = t + π t + t. So F s is bouded. Therefore fs is bouded d hece costt by Liouville. At s = / we hve π cotπ/ = d F / = / = + / = hece f. To see 9 cosider gs = siπs/πs s /. The product is bsolutely coverget so g exists for s C\Z. gs teds to s s teds to d g hs period implyig gs teds to s s teds to y iteger. The logrithmic derivtive is give by g s gs = π cotπs s + s = = s hece g is costt d sice g = we hve gs = for ll s. Propositio 4 Reflectio Formul. ΓsΓ s = π si πs. Proof. By 6 d 9 we hve So by ΓsΓ s = lim! ss + s + s! s s + s + s = lim z = π s si πs. k= + s s ΓsΓ s = Γs sγ s = π si πs.

9 THE GAMMA FUNCTION 9 Settig s = / i the reflectio formul gives Γ/ = π. Propositio 5 Duplictio Formul. We hve the followig formul ΓsΓ s + = s π / Γs Proof. The trick here is to use coveiet expressio for Γs. By 6 we hve { ΓsΓs + /! s! s+/ = lim Γs ss + s + s + s + 3 s + + } ss + s +! s = lim s = lim s = lim s { }! / +!z + + {! +! / + z/ + { }! +! / = s C Settig s = / gives C = Γ/ = π d we re doe. We fiish this sectio o the Γ fuctio with formul tht is very useful for estimtig Γs. 5. Stirlig s Formul The followig theorem chrcterises the Γ fuctio uiquely d will prove useful. Theorem Uiqueess Theorem. Let F be lytic i the right hlf-ple A = {s C : σ > }. Suppose F s + = sf s d tht F is bouded i the strip σ <. The F s = Γ i A with = F. Proof. Cosider fs = F s Γs. The equtio fs + = sfs holds i A d so we c exted f meromorphiclly to the whole ple s we did for Γ. If y poles occur these must be t the egtive itegers. Sice f = we hve lim s sfs =, hece f does t hve pole, or ythig worse, t d we c thus cotiue f lyticlly to. This gives the lytic cotiutio of f to the egtive itegers vi fs + = sfs. }

10 THE GAMMA FUNCTION Now, Γs Γσ d this is bouded for σ <. Sice F is bouded here by hypothesis, f is lso. Now cosider the regio with σ. If t = Is the f is bouded sice it s lytic here. If t > the f is bouded here sice fs = fs + /s d f is bouded for σ <. Sice fs d f s ssume the sme vlues for σ we hve tht gs = fsf s is bouded d lytic. By Liouville gs g = d hece f. Our gol is to use the uiqueess theorem to prove Γs = πs s e s e µs for s C = C\R where P x µs := s + x dx d P x = x x. We eed to show µ is lytic d tht it posseses pproprite fuctiol equtio so tht the bove represettio of Γ stisfies the fuctiol equtio. For this we eed lemm. Lemm 5 Twisted -iequlity. For s = re iθ d x we hve s + x s + x cosθ/. This gives 3 s + x s + x siδ/ whe θ π δ, < δ π sice cosθ/ siδ/. Proof. Usig cos θ = si θ/ d r + x 4rx we hve s + x = r + rx cos θ + x = r + x 4rx si θ/ r + x cos θ/ Propositio 6. µs is lytic i C. Proof. Defie Qx = x x x x. The Qt is tiderivtive of P x so cotiuous d Qx /8. We hve β β P x Qx β 4 dx = Qx s + x s + x + s + x dx α for < α < β <. Now let < δ π d ε >. The for x, s = re iθ with r > ε d θ π δ we hve by the bove lemm Qx s + x Qt si δ/ε + x 8 si δ/ε + x. Hece the itegrl Qx s + x dx α α

11 THE GAMMA FUNCTION is uiformly coverget i compct subsets of C d therefore defies lytic fuctio by Corollry. But by 4 we hve 5 µs = Qx s + x dx. 6 7 Note by 5 d, 3 we hve µs µs for s = re iθ, θ π δ, < δ π. 8 cos θ/ s 8 si δ/ s Propositio 7. For s C we hve 8 µs µs + = s + s + log. s Proof. Sice P x + = P x we hve µs + = = = µs + = µs + P x s + + x dx = P x dx = µs s + x x / s + x dx s + / s + x dx. P x + s + x + dx x x / dx s + x Theorem 3 Complex Stirlig s Formul. For s C we hve 9 Γs = πs s e s e µs. Proof. By Propositio 6 the fuctio F s = s s e s e µs is lytic i C we defie s s = e s log s where log is the pricipl brch of the logrithm. By Propositio 7 we hve F s + = s + s+/ e s e µs s+ log s+ s + = s s+/ e s e µs = sf s. Also F s is bouded i the regio A = {s C : σ > }: Clerly, e µs is bouded by 6. Writig s = σ +it = s e iθ C we hve s s e s = s σ / e θt e σ. The

12 THE GAMMA FUNCTION for s A with t we hve σ /, s t d tθ π t /. Hece, i this regio we hve s s e s 4t e π t / e s t. Hece F s is bouded i A. By the Uiqueess Theorem we must hve Γs = s s e s e µs for some. Substitutig this ito the duplictio formul gives s s / e s e µs s + / s e s / e µs+/ = s πs s / e s e µs = πs s / e s e µs. Hece + /s s e µs+µs+/ = πee µs. Now let s be rel d pproch ifiity. By 6 the expoetils ted to d sice lim x + /x x = e we hve = π. This result immeditely gives the origil form of stirlig s formul Corollry 3. As x i R we hve Γx πx x / e x. Aother cosequece is the followig. Corollry 4 Rpid decy i verticl strips. Let σ R be fixed. The s t we hve Γσ + it π t σ / e π t /. Proof. Assume t. By 9 d 6 we hve log Γσ + it = RlogΓσ + it = Rσ + it / logσ + it σ it + log π + O/t = Rσ + it /logσ + t / + i rgσ + it σ + log π + O/t = σ / log t + σ /t t rgσ + it σ + log π + O/t = σ /log t + o tπ/ t σ/t σ + log π + o = σ / log t πt/ + log π + o. Doig similr stuff for t gives the result.