Dave's Math Tables: Gamma Function

(Math | Calculus | Integrals | Special Functions | Gamma Function)

(x) = e -t t(x-1) dt	(x) = r x e -rt t (x-1) dt
(x) = 2 e (-t^2) t (2x-1) dt	( (x)(y) ) / ( (x) + (y) ) = Beta(x,y)
(x+1) = x (x)	(x+1) = x!
(0+) = +	(1/2) = (PI)
(z) (1-z) = PI csc(PI z)	'(1) = - gamma (Euler's constant = 0.577215...)
'(x) = (x) lim (n-->) [ ln(n) - (k=0..n-1) 1/(x+k) ]

Function Expansions

(x+1) = lim (k-->) kx 1*2*3*..*k (x+1)(x+2)*..*(x+k)

Stirling's Asymptotic Series (x+1) = (2PIx) xx e -x { 1 + 1/(12x) + 1/(288 x2) - 139/(51840 x3) - 571/(2488320 x4) + 163879/(209018880 x5) + 5246819/(75246796800 x6) - 534703531/(902961561600 x7) - ...}

Incomplete Gamma Functions:
(x, a) = e ^-t t ^(x-1) dt
(x, a) = e ^-t t ^(x-1) dt

(x, a) + (x, a) = (x)