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 Gamma Function/b> (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) - ...}
(this is only an approximation, see note.)

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

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