The Gamma fuctio Marco Bovii October 9, 2 Gamma fuctio The Euler Gamma fuctio is defied as Γ() It is easy to show that Γ() satisfy the recursio relatio ideed, itegratig by parts, dt e t t. () Γ( + ) Γ() : (2) Γ( + ) dt e t t e t t + dt e t t Γ(). (3) For iteger eq. (2) is the recursio relatio of the factorial, ad thus we have Γ( + ) ; (4) because i additio Γ() (easly derived from the defiitio), we have the idetificatio Γ( + ), (5) ad i this sese the Gamma fuctio is a complex extesio of the factorial. The sequece of Gamma fuctio computed i all half-itegers ca be obtaied usig subsequetly the recursio relatio (2) ad kowig that ( Γ 2) π (6) that is easy to compute: ( Γ 2) 2 2 dt e t t 2 dx e x2 π 2 π. (7)
5 5 2 2 4 5 Figure : Re Γ(x) for real x.. Aalytical structure First, from the defiitio (), we see that Γ( ) Γ(), (8) that is to say that Gamma is a real fuctio, ad i particular Im Γ(x) for x R. Ivertig eq. (2) we have Γ() Γ( + ), (9) ad whe it diverges (because Γ() is fiite). It is evidet that correspods to a simple pole of order ad residue : ideed Res Γ() Γ() Γ( + ). () Formally, we ca discover that the Gamma fuctio has simple poles i all egative itegers simply iteratig the recursio (2): Γ( ) Γ( + )... ( ) Γ() ( N). () This brigs to the coclusio that Γ( ) has a simple pole of order ad residue ( ). More rigorously, this ca be show i the usual way: Res Γ( ) Γ( )... Γ( + ) 2
( )( + ) ( ) Γ() ( ) ( N). (2) I all the complex plae (except the egative real axis) the Gamma fuctio is well defied. Fo o-iteger egative real values the Gamma fuctio ca be aalytically cotiued (as we have see for example for half-itegers, positives ad egatives). We ca prove these results i a simpler way. Startig from the defiitio () we rewrite Γ() dt e t t + dt e t t dt e t t + dt e t t + dt t ( ) ( ) + where the first itegral is a aalytic fuctio of ad the secod term gives the positio of the poles ad their residues. Moreover, the Gamma fuctio has a essetial sigularity to complex ifiity, because Γ ( ) has a o-defied limit for. This is to say that the Gamma fuctio is ot well defied i the compactified complex plae. Now we wat to show that ear a pole i oe has the expasio Γ( ) ( ) [ + ψ( + ) + O() ] where ψ() is the logarithmic derivative of the Gamma fuctio, defied i (4); ote that ψ( + ) has a explicit expressio give i eq. (44). First, ote that where we have defied Γ( ) Γ( + )... Γ( + + ) ( )( + ) ( ) ( + ) ( + ) G(, ) t (3) (4) G(, ) (5) Γ( + + ) ( ) [ ( 2 2 ) ( 2 Γ( + + ) + O( 2 ) () 2 ) ]. (6) By Taylor expadig G(, ) aroud i (5) we get that is the claimed result. Γ( ) ( ) [ G(, ) + G (, ) + O( 2 ) ] [ + Γ ( + ) + O()] 3, (7)
.2 Alterative defiitios There are some alterative defiitios of the Gamma fuctio. Oe, due to Euler, is Γ() ( + ) + ( + ) ( + ) ( + ). (8) It is easy to prove (usig the expressio with the limit) that with this defiitio the recursio relatio (2) is satisfied, ad also that for gives Γ(). Aother oe, due to Weierstrass, is Γ() e γ e / + e γ lim [ ( exp + 2 +... + ( + ) ( + ) )], (9) where γ is the Euler-Mascheroi umber defied i Sectio 2.2 The equivalece to the Euler defiitio ca be see by takig the logarithm of this two forms: the first gives ad the secod gives log Γ() log + log Γ() log γ + [ ( log + ) ( log + )] [ ( log + )] (2). (2) Comparig this two expressios we ote that they are equal because of eq. (48). Note that this two expressio are useful i order to umerically approximate log Γ(), by trucatig the series to a certai value of..3 Euler reflectio formula A useful formula is Γ() Γ( ) π si(π) called the Euler reflectio formula. To demostrate it, cosider the alterative defiitio (8) ad ote that where we have used Γ() Γ( ) Γ() Γ( ) ( ) ( + ) ( ) ( + ) ( + ) ( si(π) π 2 2 ) (22) (23) ( ) si(π) π 2 2. (24) 4
.4 Stirlig approximatio The Stirlig approximatio for the factorial is 2π e (25) which is valid for large. To demostrate it, we perform a saddle poit approximatio to the itegral (): Γ( + ) dt e t t t+ log t dt e ow we expad the expoet ear its maximum ad get that is the claimed result..5 Derivative recursio relatio dt exp ( + log ) 2 t2 e dt e 2 t2 Cosider the Taylor expasio ear of Γ( + + ): 2π e (26) Γ( + + ) k Γ (k) ( + ) k ; (27) usig eq. (2) this is equal to ( + ) Γ( + ), that expaded is ( + ) Γ( + ) ( + ) k Γ( ) + Γ (k) ( ) Comparig the two expasios, we get the equality (k > ) k k k [ Γ (k) ( ) + k Γ (k ) ( ) ]. (28) Γ (k) ( + ) Γ (k) ( ) + k Γ (k ) ( ). (29).6 Iverse Gamma fuctio We ca defie a iverse Gamma fuctio () Γ() (3) 5
2..5..5 2 2 4 6.5. Figure 2: Re (x) for real x. which is a etire fuctio (aalytical i all the complex plae). The recursio relatio (2) becomes () ( + ). (3) We are able to produce a itegral represetatio of the fuctio: cosider the Laplace trasform of the fuctio t with respect to t: dt e st t s Γ() (32) ad the we have that the iverse Laplace traform of s is c+i ds e ts s () t (33) 2πi c i where c has to be choose to the right of all the sigularities of s, that is c >. Now, by choosig t, we get () c+i ds e s s, (34) 2πi c i where the itegratio path ca be deformed to be a cotour of the egative real axis. Note that for positive itegers + the brach cut vaishes ad oly a pole i s remais, ad the the path ca be closed aroud s to obtai 2πi that ca be also easly demostrated usig the residue theorem. Now cosider the Taylor expasio ear of ( + ): ( + ) k ds e s s, (35) (k) ( ) usig eq. (3) this is equal to ( + ) ( + + ), that expaded is ( + ) ( + + ) ( + ) k (k) ( + ) 6 k k ; (36)
k [ ( + ) + (k) ( + ) + k (k ) ( + ) ]. (37) Comparig the two expasios, we get the equality (k ) k (k) ( ) (k) ( + ) + k (k ) ( + ) (38) ad, for, (k) () k (k ) (). (39).6. Saddle poit approximatio The aalogous of the Stirlig approximatio ca be doe for the () fuctio. The result is: () 2π e. (4) 7
2 Gamma related fuctios 2. Log derivative ψ() Defie the logarithmic derivative of the Gamma fuctio as follows Form the recursio relatio (2) it follows that ψ() d log Γ(). (4) d ψ( + ) ψ() +, (42) as we ca easly demostrate: ψ( + ) d log Γ( + ) d d log [ Γ()] d + ψ() (43) (ote that the same result ca be obtaied from eq. (29) for k ). For a positive iteger, we ca iterate the recursio (42) to obtai ψ( + ) ψ() + + 2 + 3 +... +, (44) ad the value of ψ() is ψ() γ (45) where γ is the Euler-Mascheroi umber, as demostrated i Sectio 2.2. The fuctios ψ () ψ () () d+ log Γ() (46) d+ are also defied. Iteratig the derivative i (42) we obtai the recursio relatio 2.2 Euler-Mascheroi umber The Euler-Mascheroi umber is defied as ψ ( + ) ψ () + ( ) +. (47) γ [ ( log + )], (48) ad its value is γ.57726.... (49) 8
Note that ( log + ) N N N ( log + ) N [ ] log( + ) log N log( + N) N log N, (5) where we oted that the sum is a telescopic sum ad that i the limit N log( + N) ad log N are equivalet. Thus we ca rewrite eq. (48) as γ N N log N. (5) Now we prove that ψ() γ. To do this, let us cosider eq. (2), ad take its derivative: ψ() [ ( + log + ) ] +. (52) For we get ψ() + i which we recogise the defiitio (48). 2.3 Geeralied Gamma fuctios [ ( log + ) ] + [ ( log + )] You ca also defie a icomplete Gamma fuctio (or plica fuctio) ad a trucated Gamma fuctio Γ(, a) γ(, a) a a, (53) dt e t t (54) dt e t t, (55) that have a brach cut o the egative real axis i the a complex plae. Obviously Γ() Γ(, a) + γ(, a). (56) 9
For iteger k +, itegratig repeatedly by parts we get 2.4 Beta fuctio The Beta fuctio is defied as Γ(k +, a) e a k γ(k +, a) e a a (57) k a. (58) B(a, b) dx x a ( x) b Γ(a)Γ(b) Γ(a + b). (59) To show the last equality, cosider the product Γ(a)Γ(b) 4 dt e t t a du e u u b dx 2π 2 Γ(a + b) Γ(a + b) dx e x2 x 2a dy e y2 y 2b dy e (x2 +y 2) x 2a y 2b dθ cos 2a θ si 2b θ π/2 dθ cos 2a θ si 2b θ dr e r2 r 2a+2b dx x a ( x) b. (6)