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:

\begin{eqnarray*} \gamma (a,x) &=& \int_0^x e^{-t} t^{a-1} \, dt \\ \Gamma (a,x) &=& \int_x^\infty e^{-t} t^{a-1} \, dt = \Gamma(a) - \gamma (a,x) \\ P (a,x) &=& {1 \over \Gamma (a)} \int_0^x e^{-t} t^{a-1} \, dt = {\gamma (a,x) \over \Gamma(a)} \\ \gamma^* (a,x) &=& {x^{-a} \over \Gamma(a)} \int_0^x e^{-t} t^{a-1} \, dt = {\gamma (a,x) \over x^a \Gamma(a)} \\ I (a,x) &=& {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)} \\ C (a,x) &=& \int_x^\infty t^{a-1} \cos t \, dt \\ S (a,x) &=& \int_x^\infty t^{a-1} \sin t \, dt \\ E_n (x) &=& \int_1^\infty e^{-xt} t^{-n} \, dt = x^{n-1} \Gamma (1-n) - x^{n-1} \gamma (1-n,x) \\ \alpha_n (x) &=& \int_1^\infty e^{-xt} t^n \, dt = x^{-n-1} \Gamma (1+n) - x^{-n-1} \gamma (1+n,x) \\ \end{eqnarray*}For convenience of translating notations, these variants have been expressed in terms of $\gamma$ .