# §8.14 Integrals

 8.14.1 $\int_{0}^{\infty}e^{-ax}\frac{\mathop{\gamma\/}\nolimits\!\left(b,x\right)}{% \mathop{\Gamma\/}\nolimits\!\left(b\right)}dx=\frac{(1+a)^{-b}}{a},$ $\realpart{a}>0$, $\realpart{b}>-1$,
 8.14.2 $\int_{0}^{\infty}e^{-ax}\mathop{\Gamma\/}\nolimits\!\left(b,x\right)dx=\mathop% {\Gamma\/}\nolimits\!\left(b\right)\frac{1-(1+a)^{-b}}{a},$ $\realpart{a}>-1$, $\realpart{b}>-1$.

In (8.14.1) and (8.14.2) limiting values are used when $b=0$.

 8.14.3 $\int_{0}^{\infty}x^{a-1}\mathop{\gamma\/}\nolimits\!\left(b,x\right)dx=-\frac{% \mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a},$ $\realpart{a}<0$, $\realpart{(a+b)}>0$,
 8.14.4 $\int_{0}^{\infty}x^{a-1}\mathop{\Gamma\/}\nolimits\!\left(b,x\right)dx=\frac{% \mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a},$ $\realpart{a}>0$, $\realpart{(a+b)}>0$,
 8.14.5 $\int_{0}^{\infty}x^{a-1}e^{-sx}\mathop{\gamma\/}\nolimits\!\left(b,x\right)dx=% \frac{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{b(1+s)^{a+b}}\*\mathop{F\/% }\nolimits\!\left(1,a+b;1+b;1/(1+s)\right),$ $\realpart{s}>0$, $\realpart{(a+b)}>0$,
 8.14.6 $\int_{0}^{\infty}x^{a-1}e^{-sx}\mathop{\Gamma\/}\nolimits\!\left(b,x\right)dx=% \frac{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a(1+s)^{a+b}}\*\mathop{F\/% }\nolimits\!\left(1,a+b;1+a;s/(1+s)\right),$ $\realpart{s}>-1$, $\realpart{(a+b)}>0$, $\realpart{a}>0$.

For the hypergeometric function $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ see §15.2(i).

For additional integrals see Apelblat (1983, §8.2), Erdélyi et al. (1953b, §9.3), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §6.45), Marichev (1983, pp.189–190), Oberhettinger (1972, pp. 68–69), Prudnikov et al. (1986b, §§1.2, 2.10), and Prudnikov et al. (1992a, §3.10).