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:

$\gamma (a,x)$$=$${\displaystyle {\int }_0^x}{e}^{-t}{t}^{a-1}𝑑t$

$\mathrm{\Gamma }(a,x)$$=$${\displaystyle {\int }_x^{\mathrm{\infty }}}{e}^{-t}{t}^{a-1}𝑑t=\mathrm{\Gamma }(a)-\gamma (a,x)$

$P(a,x)$$=$${\displaystyle \frac{1}{\mathrm{\Gamma }(a)}}{\displaystyle {\int }_0^x}{e}^{-t}{t}^{a-1}𝑑t={\displaystyle \frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}}$

${\gamma }^{*}(a,x)$$=$${\displaystyle \frac{x^{-a}}{\mathrm{\Gamma }(a)}}{\displaystyle {\int }_0^x}{e}^{-t}{t}^{a-1}𝑑t={\displaystyle \frac{\gamma (a,x)}{x^a\mathrm{\Gamma }(a)}}$

$I(a,x)$$=$${\displaystyle \frac{1}{\mathrm{\Gamma }(a+1)}}{\displaystyle {\int }_0^{x\sqrt{a+1}}}{e}^{-t}{t}^a𝑑t={\displaystyle \frac{\gamma (a+1,x\sqrt{a+1})}{\mathrm{\Gamma }(a+1)}}$

$C(a,x)$$=$${\displaystyle {\int }_x^{\mathrm{\infty }}}{t}^{a-1}\cos tdt$

$S(a,x)$$=$${\displaystyle {\int }_x^{\mathrm{\infty }}}{t}^{a-1}\sin tdt$

$E_n(x)$$=$${\displaystyle {\int }_1^{\mathrm{\infty }}}{e}^{-xt}{t}^{-n}𝑑t=x^{n-1}\mathrm{\Gamma }(1-n)-x^{n-1}\gamma (1-n,x)$

${\alpha }_n(x)$$=$${\displaystyle {\int }_1^{\mathrm{\infty }}}{e}^{-xt}{t}^n𝑑t=x^{-n-1}\mathrm{\Gamma }(1+n)-x^{-n-1}\gamma (1+n,x)$

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