# §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 $\realpart{a}>0$ and $\realpart{b}>0$.

# Euler’s Beta Integral

 5.12.1 $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1% }dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits% \!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}.$ Defines: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$: beta function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: 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
 5.12.2 $\int_{0}^{\pi/2}{\mathop{\sin\/}\nolimits^{2a-1}}\theta{\mathop{\cos\/}% \nolimits^{2b-1}}\theta d\theta=\tfrac{1}{2}\mathop{\mathrm{B}\/}\nolimits\!% \left(a,b\right).$
 5.12.3 $\int_{0}^{\infty}\frac{t^{a-1}dt}{(1+t)^{a+b}}=\mathop{\mathrm{B}\/}\nolimits% \!\left(a,b\right).$
 5.12.4 $\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}dt=\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right)(1+z)^{-a}z^{-b},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$.
 5.12.5 $\int_{0}^{\pi/2}(\mathop{\cos\/}\nolimits t)^{a-1}\mathop{\cos\/}\nolimits\!% \left(bt\right)dt=\frac{\pi}{2^{a}}\frac{1}{a\mathop{\mathrm{B}\/}\nolimits\!% \left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$.
 5.12.6 $\int_{0}^{\pi}(\mathop{\sin\/}\nolimits t)^{a-1}e^{ibt}dt=\frac{\pi}{2^{a-1}}% \frac{e^{i\pi b/2}}{a\mathop{\mathrm{B}\/}\nolimits\!\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$.
 5.12.7 $\int_{0}^{\infty}\frac{\mathop{\cosh\/}\nolimits\!\left(2bt\right)}{(\mathop{% \cosh\/}\nolimits t)^{2a}}dt=4^{a-1}\mathop{\mathrm{B}\/}\nolimits\!\left(a+b,% a-b\right),$ $\realpart{a}>|\realpart{b}|$.
 5.12.8 ${\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{dt}{(w+it)^{a}(z-it)^{b}}=\frac{(w% +z)^{1-a-b}}{(a+b-1)\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)}},$ $\realpart{(a+b)}>1$, $\realpart{w}>0$, $\realpart{z}>0$.

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 ${\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}dt=\frac{1}{b% \mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)}},$ $0, $\realpart{(a+b)}>0$.
 5.12.10 ${\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}dt=\frac{\mathop{\sin\/}% \nolimits\!\left(\pi b\right)}{\pi}\mathop{\mathrm{B}\/}\nolimits\!\left(a,b% \right)},$ $\realpart{a}>0$,

with the contour as shown in Figure 5.12.1.

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

when $\realpart{b}>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. See Figure 5.12.2.

# Pochhammer’s Integral

When $a,b\in\Complex$

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

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.