§ 5.13. Integrals

Notes:: For (5.13.1) see Paris and Kaminski (2001, p. 96); (5.13.2) follows by taking $c=0$ , $b=a$ , $s=it$ and $z=e^{{i(\pi-2b)}}$ . For (5.13.3)–(5.13.4) see Titchmarsh (1986a, pp. 188 and 194), and for (5.13.5) see Andrews et al. (1999, p. 152).
Permalink:: http://dlmf.nist.gov/5.13

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

5.13.1 ${\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\Gamma\!\left(s+a\right)\Gamma\!\left(b-s\right)z^{{-s}}ds=\frac{\Gamma\!\left(a+b\right)z^{a}}{(1+z)^{{a+b}}}},$ $\realpart{(a+b)}>0$ , $-\realpart{a}<c<\realpart{b}$ , $|\mathrm{ph}z|<\pi$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $s$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13, §5.13, §5.19(ii)
Permalink:: http://dlmf.nist.gov/5.13.E1
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5.13.2 ${\frac{1}{2\pi}\int _{{-\infty}}^{\infty}|\Gamma\!\left(a+it\right)|^{2}e^{{(2b-\pi)t}}dt=\frac{\Gamma\!\left(2a\right)}{(2\sin b)^{{2a}}}},$ $a>0$ , $0<b<\pi$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E2
Encodings:: TeX, pMathML, png

¶ Barnes' Beta Integral

5.13.3 $\frac{1}{2\pi}\int _{{-\infty}}^{\infty}\Gamma\!\left(a+it\right)\Gamma\!\left(b+it\right)\Gamma\!\left(c-it\right)\Gamma\!\left(d-it\right)dt=\frac{\Gamma\!\left(a+c\right)\Gamma\!\left(a+d\right)\Gamma\!\left(b+c\right)\Gamma\!\left(b+d\right)}{\Gamma\!\left(a+b+c+d\right)},$ $\realpart{a},\realpart{b},\realpart{c},\realpart{d}>0$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E3
Encodings:: TeX, pMathML, png

¶ Ramanujan's Beta Integral

5.13.4 $\int _{{-\infty}}^{\infty}\frac{dt}{\Gamma\!\left(a+t\right)\Gamma\!\left(b+t\right)\Gamma\!\left(c-t\right)\Gamma\!\left(d-t\right)}=\frac{\Gamma\!\left(a+b+c+d-3\right)}{\Gamma\!\left(a+c-1\right)\Gamma\!\left(a+d-1\right)\Gamma\!\left(b+c-1\right)\Gamma\!\left(b+d-1\right)},$ $\realpart{(a+b+c+d)}>3$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E4
Encodings:: TeX, pMathML, png

¶ de Branges-Wilson Beta Integral

5.13.5 $\frac{1}{4\pi}\int _{{-\infty}}^{\infty}\frac{\prod _{{k=1}}^{4}\Gamma\!\left(a_{k}+it\right)\Gamma\!\left(a_{k}-it\right)}{\Gamma\!\left(2it\right)\Gamma\!\left(-2it\right)}dt=\frac{\prod _{{1\le j<k\le 4}}\Gamma\!\left(a_{j}+a_{k}\right)}{\Gamma\!\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},$ $\realpart{(a_{k})}>0$ , $k=1,2,3,4$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $j$ : nonnegative integer, $k$ : nonnegative integer and $a$ : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E5
Encodings:: TeX, pMathML, png

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. (1986, pp. 57–64), Prudnikov et al. (1992a, pp. 127–130), and Prudnikov et al. (1992b, pp. 113–123).