incomplete gamma function

The incomplete gamma functionDlmfDlmfDlmfDlmfMathworldPlanetmath is defined as the indefinite integral of the integrand of gamma integralDlmfPlanetmath. There are several definitions which differ in details of normalization and constant of integration:

γ(a,x) = 0xe-tta-1𝑑t
Γ(a,x) = xe-tta-1𝑑t=Γ(a)-γ(a,x)
P(a,x) = 1Γ(a)0xe-tta-1𝑑t=γ(a,x)Γ(a)
γ*(a,x) = x-aΓ(a)0xe-tta-1𝑑t=γ(a,x)xaΓ(a)
I(a,x) = 1Γ(a+1)0xa+1e-tta𝑑t=γ(a+1,xa+1)Γ(a+1)
C(a,x) = xta-1costdt
S(a,x) = xta-1sintdt
En(x) = 1e-xtt-n𝑑t=xn-1Γ(1-n)-xn-1γ(1-n,x)
αn(x) = 1e-xttn𝑑t=x-n-1Γ(1+n)-x-n-1γ(1+n,x)

For convenience of translating notations, these variants have been expressed in terms of γ.

Title incomplete gamma function
Canonical name IncompleteGammaFunction
Date of creation 2013-03-22 15:36:48
Last modified on 2013-03-22 15:36:48
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 30D30
Classification msc 33B15
Related topic SineIntegralInInfinity