# incomplete gamma function

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

 $\displaystyle\gamma(a,x)$ $\displaystyle=$ $\displaystyle\int_{0}^{x}e^{-t}t^{a-1}\,dt$ $\displaystyle\Gamma(a,x)$ $\displaystyle=$ $\displaystyle\int_{x}^{\infty}e^{-t}t^{a-1}\,dt=\Gamma(a)-\gamma(a,x)$ $\displaystyle P(a,x)$ $\displaystyle=$ $\displaystyle{1\over\Gamma(a)}\int_{0}^{x}e^{-t}t^{a-1}\,dt={\gamma(a,x)\over% \Gamma(a)}$ $\displaystyle\gamma^{*}(a,x)$ $\displaystyle=$ $\displaystyle{x^{-a}\over\Gamma(a)}\int_{0}^{x}e^{-t}t^{a-1}\,dt={\gamma(a,x)% \over x^{a}\Gamma(a)}$ $\displaystyle I(a,x)$ $\displaystyle=$ $\displaystyle{1\over\Gamma(a+1)}\int_{0}^{x\sqrt{a+1}}e^{-t}t^{a}\,dt={\gamma(% a+1,x\sqrt{a+1})\over\Gamma(a+1)}$ $\displaystyle C(a,x)$ $\displaystyle=$ $\displaystyle\int_{x}^{\infty}t^{a-1}\cos t\,dt$ $\displaystyle S(a,x)$ $\displaystyle=$ $\displaystyle\int_{x}^{\infty}t^{a-1}\sin t\,dt$ $\displaystyle E_{n}(x)$ $\displaystyle=$ $\displaystyle\int_{1}^{\infty}e^{-xt}t^{-n}\,dt=x^{n-1}\Gamma(1-n)-x^{n-1}% \gamma(1-n,x)$ $\displaystyle\alpha_{n}(x)$ $\displaystyle=$ $\displaystyle\int_{1}^{\infty}e^{-xt}t^{n}\,dt=x^{-n-1}\Gamma(1+n)-x^{-n-1}% \gamma(1+n,x)$

For convenience of translating notations, these variants have been expressed in terms of $\gamma$.

Title incomplete gamma function IncompleteGammaFunction 2013-03-22 15:36:48 2013-03-22 15:36:48 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Definition msc 30D30 msc 33B15 SineIntegralInInfinity