Gamma Function

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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...)

Gamma '(x) = Gamma (x) lim (n--> inf ) [ ln(n) - sum _(k=0..n-1) 1/(x+k) ]

Function Expansions

Gamma (x+1) = lim _(k-->) k^x 1*2*3*..*k
(x+1)(x+2)*..*(x+k)

Stirling's Asymptotic Series
Gamma (x+1) = sqrt (2PIx) x^x 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)