# incomplete gamma function

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

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