# §5.13 Integrals

In (5.13.1) the integration path is a straight line parallel to the imaginary axis.

 5.13.1 ${\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\Gamma\left% (b-s\right)z^{-s}\,\mathrm{d}s=\frac{\Gamma\left(a+b\right)z^{a}}{(1+z)^{a+b}}},$ $\Re\left(a+b\right)>0$, $-\Re a, $|\operatorname{ph}z|<\pi$.
 5.13.2 ${\frac{1}{2\pi}\int_{-\infty}^{\infty}|\Gamma\left(a+it\right)|^{2}e^{(2b-\pi)% t}\,\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b)^{2a}}},$ $a>0$, $0.

## Barnes’ Beta Integral

 5.13.3 $\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a+it\right)\Gamma\left(b+it% \right)\Gamma\left(c-it\right)\Gamma\left(d-it\right)\,\mathrm{d}t=\frac{% \Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(% b+d\right)}{\Gamma\left(a+b+c+d\right)},$ $\Re a,\Re b,\Re c,\Re d>0$.

## Ramanujan’s Beta Integral

 5.13.4 $\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{\Gamma\left(a+t\right)\Gamma\left(% b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t\right)}=\frac{\Gamma\left(a+b+% c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma\left(a+d-1\right)\Gamma\left(b+c-% 1\right)\Gamma\left(b+d-1\right)},$ $\Re\left(a+b+c+d\right)>3$.

## de Branges–Wilson Beta Integral

 5.13.5 $\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\Gamma\left(a_{k}+it% \right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2it\right)\Gamma\left(-2it% \right)}\,\mathrm{d}t=\frac{\prod_{1\leq j $\Re\left(a_{k}\right)>0$, $k=1,2,3,4$.

For compendia of integrals of gamma functions see Apelblat (1983, pp. 124–127 and 129–130), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, pp. 644–652), Oberhettinger (1974, pp. 191–204), Oberhettinger and Badii (1973, pp. 307–316), Prudnikov et al. (1986b, pp. 57–64), Prudnikov et al. (1992a, pp. 127–130), and Prudnikov et al. (1992b, pp. 113–123).