5 Gamma Function Properties 5.12 Beta Function 5.14 Multidimensional Integrals

§5.13 Integrals

Notes:: For (5.13.1) see Paris and Kaminski (2001, p. 96); (5.13.2) follows by taking , , 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).
Keywords:: beta function, gamma function, integrals
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 ${\frac{1}{2\pi i}\int_{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!% \left(s+a\right)\mathop{\Gamma\/}\nolimits\!\left(b-s\right)z^{{-s}}ds=\frac{% \mathop{\Gamma\/}\nolimits\!\left(a+b\right)z^{a}}{(1+z)^{{a+b}}}},$ $\realpart{(a+b)}>0$ , $-\realpart{a}<c<\realpart{b}$ , $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, : real or complex variable, : complex variable, : real or complex variable and : 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 ${\frac{1}{2\pi}\int_{{-\infty}}^{\infty}|\mathop{\Gamma\/}\nolimits\!\left(a+% it\right)|^{2}e^{{(2b-\pi)t}}dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(2a% \right)}{(2\mathop{\sin\/}\nolimits b)^{{2a}}}},$

, $0<b<\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex variable and : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E2
Encodings:: TeX, pMML, png

¶ Barnes’ Beta Integral

Keywords:: Barnes’ beta integral

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real or complex variable and : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E3
Encodings:: TeX, pMML, png

¶ Ramanujan’s Beta Integral

Keywords:: Ramanujan’s beta integral

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real or complex variable and : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E4
Encodings:: TeX, pMML, png

¶ de Branges–Wilson Beta Integral

Keywords:: beta integrals, de Branges–Wilson beta integral

5.13.5 $\frac{1}{4\pi}\int_{{-\infty}}^{\infty}\frac{\prod_{{k=1}}^{4}\mathop{\Gamma\/% }\nolimits\!\left(a_{k}+it\right)\mathop{\Gamma\/}\nolimits\!\left(a_{k}-it% \right)}{\mathop{\Gamma\/}\nolimits\!\left(2it\right)\mathop{\Gamma\/}% \nolimits\!\left(-2it\right)}dt=\frac{\prod_{{1\leq j<k\leq 4}}\mathop{\Gamma% \/}\nolimits\!\left(a_{j}+a_{k}\right)}{\mathop{\Gamma\/}\nolimits\!\left(a_{1% }+a_{2}+a_{3}+a_{4}\right)},$ $\realpart{(a_{k})}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : nonnegative integer, : nonnegative integer and : real or complex variable
Referenced by:: §5.13
Permalink:: http://dlmf.nist.gov/5.13.E5
Encodings:: TeX, pMML, 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. (1986b, pp. 57–64), Prudnikov et al. (1992a, pp. 127–130), and Prudnikov et al. (1992b, pp. 113–123).

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