# §8.6 Integral Representations

## §8.6(i) Integrals Along the Real Line

For the Bessel function $J_{\nu}\left(z\right)$ and modified Bessel function $K_{\nu}\left(z\right)$, see §§10.2(ii) and 10.25(ii).

 8.6.1 $\displaystyle\gamma\left(a,z\right)$ $\displaystyle=\frac{z^{a}}{\sin\left(\pi a\right)}\int_{0}^{\pi}e^{z\cos t}% \cos\left(at+z\sin t\right)\mathrm{d}t,$ $a\notin\mathbb{Z}$, 8.6.2 $\displaystyle\gamma\left(a,z\right)$ $\displaystyle=z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}J_{a}% \left(2\sqrt{zt}\right)\mathrm{d}t,$ $\Re a>0$. 8.6.3 $\displaystyle\gamma\left(a,z\right)$ $\displaystyle=z^{a}\int_{0}^{\infty}\exp\left(-at-ze^{-t}\right)\mathrm{d}t,$ $\Re a>0$.
 8.6.4 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=\frac{z^{a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}\frac% {t^{-a}e^{-t}}{z+t}\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$, $\Re a<1$, 8.6.5 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\mathrm{d% }t,$ $\Re z>0$, 8.6.6 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=\frac{2z^{\frac{1}{2}a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{% \infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(2\sqrt{zt}\right)\mathrm{d}t,$ $\Re a<1$, 8.6.7 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=z^{a}\int_{0}^{\infty}\exp\left(at-ze^{t}\right)\mathrm{d}t,$ $\Re z>0$.

## §8.6(ii) Contour Integrals

 8.6.8 $\gamma\left(a,z\right)=\frac{-\mathrm{i}z^{a}}{2\sin\left(\pi a\right)}\int_{-% 1}^{(0+)}t^{a-1}e^{zt}\mathrm{d}t,$ $z\neq 0$, $a\notin\mathbb{Z}$;

$t^{a-1}$ takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path.

 8.6.9 $\Gamma\left(-a,ze^{\pm\pi i}\right)=\frac{e^{z}e^{\mp\pi\mathrm{i}a}}{\Gamma% \left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\mathrm{d}t,$ $\Re z>0$, $\Re a>-1$,

where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken.

### Mellin–Barnes Integrals

In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that it separates the poles of the gamma function from the other pole in the integrand, in the case of (8.6.10) and (8.6.11), and from the poles at $s=0,1,2,\dots$ in the case of (8.6.12).

 8.6.10 $\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{% \Gamma\left(s\right)}{a-s}z^{a-s}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, $a\neq 0,-1,-2,\dots$,
 8.6.11 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a% \right)\frac{z^{-s}}{s}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, 8.6.12 $\displaystyle\Gamma\left(a,z\right)$ $\displaystyle=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}{2\pi i}% \int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{\sin% \left(\pi s\right)}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{3}{2}\pi$, $a\neq 1,2,3,\dots$.

## §8.6(iii) Compendia

For collections of integral representations of $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).