# §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 in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$.

 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).