# §5.12 Beta Function

In this section all fractional powers have their principal values, except where noted otherwise. In (5.12.1)–(5.12.4) it is assumed $\Re a>0$ and $\Re b>0$.

## Euler’s Beta Integral

 5.12.1 $\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}.$ ⓘ Defines: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$: beta function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $a$: real or complex variable and $b$: real or complex variable A&S Ref: 6.2.1 and 6.2.2 Referenced by: §10.22(ii), §10.43(iii), §19.20(i), §19.20(iv), §2.6(iii), §5.12, §5.12, §7.7(ii) Permalink: http://dlmf.nist.gov/5.12.E1 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5
 5.12.2 $\int_{0}^{\pi/2}{\sin}^{2a-1}\theta{\cos}^{2b-1}\theta\mathrm{d}\theta=\tfrac{% 1}{2}\mathrm{B}\left(a,b\right).$
 5.12.3 $\int_{0}^{\infty}\frac{t^{a-1}\mathrm{d}t}{(1+t)^{a+b}}=\mathrm{B}\left(a,b% \right).$ ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$: beta function, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $a$: real or complex variable and $b$: real or complex variable A&S Ref: 6.2.1 and 6.2.2 Referenced by: §5.12 Permalink: http://dlmf.nist.gov/5.12.E3 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5
 5.12.4 $\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\mathrm{d}t=\mathrm{B}\left(% a,b\right)(1+z)^{-a}z^{-b},$ $|\operatorname{ph}z|<\pi$.
 5.12.5 $\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)\mathrm{d}t=\frac{\pi}{2^{a}}% \frac{1}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\Re a>0$.
 5.12.6 $\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\mathrm{d}t=\frac{\pi}{2^{a-1}}\frac{e^{i% \pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\Re a>0$.
 5.12.7 $\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(\cosh t)^{2a}}\mathrm{d}t=4^{a-% 1}\mathrm{B}\left(a+b,a-b\right),$ $\Re a>|\Re b|$.
 5.12.8 ${\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\mathrm{d}t}{(w+it)^{a}(z-it)^{b}}% =\frac{(w+z)^{1-a-b}}{(a+b-1)\mathrm{B}\left(a,b\right)}},$ $\Re\left(a+b\right)>1$, $\Re w>0$, $\Re z>0$.

In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity.

 5.12.9 ${\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}\mathrm{d}t=% \frac{1}{b\mathrm{B}\left(a,b\right)}},$ $0, $\Re\left(a+b\right)>0$.
 5.12.10 ${\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}\mathrm{d}t=\frac{\sin\left(% \pi b\right)}{\pi}\mathrm{B}\left(a,b\right)},$ $\Re a>0$,

with the contour as shown in Figure 5.12.1. Figure 5.12.1: t-plane. Contour for first loop integral for the beta function. Magnify

In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning.

 5.12.11 $\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\mathrm{d}t=% \mathrm{B}\left(a,b\right),$

when $\Re b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. See Figure 5.12.2. Figure 5.12.2: t-plane. Contour for second loop integral for the beta function. Magnify

## Pochhammer’s Integral

When $a,b\in\mathbb{C}$

 5.12.12 $\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{d}t=-4e^{\pi i(a+b)}\sin% \left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}\left(a,b\right),$

where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. It can always be deformed into the contour shown in Figure 5.12.3. Figure 5.12.3: t-plane. Contour for Pochhammer’s integral. Magnify