§ 5.12. Beta Function

Notes:: For (5.12.1)–(5.12.4) see Carlson (1977, pp. 60 and 70). For (5.12.5)–(5.12.6) and (5.12.8) see Nielsen (1906, §64). For (5.12.7), (5.12.9), (5.12.10), and (5.12.12) see Temme (1996, pp. 74–75) and Olver (1997b, p. 38). (An error in Ex.3.13 of Temme (1996) is corrected here.) (5.12.11) follows from (5.12.3).
Permalink:: http://dlmf.nist.gov/5.12

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 $\mathrm{B}\!\left(a,b\right)=\int _{0}^{1}t^{{a-1}}(1-t)^{{b-1}}dt=\frac{\Gamma\!\left(a\right)\Gamma\!\left(b\right)}{\Gamma\!\left(a+b\right)}.$

Defines:: $\mathrm{B}\!\left(a,b\right)$ : Beta function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $a$ : real or complex variable and $b$ : real or complex variable
A&S Ref:: 6.2.1 and 6.2.2
Referenced by:: §2.6(iii), §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E1
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5.12.2 $\int _{0}^{{\pi/2}}{\sin^{{2a-1}}}\theta{\cos^{{2b-1}}}\theta d\theta=\tfrac{1}{2}\mathrm{B}\!\left(a,b\right).$

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
A&S Ref:: 6.2.1 and 6.2.2
Permalink:: http://dlmf.nist.gov/5.12.E2
Encodings:: TeX, pMathML, png

5.12.3 $\int _{0}^{\infty}\frac{t^{{a-1}}dt}{(1+t)^{{a+b}}}=\mathrm{B}\!\left(a,b\right).$

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : 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, pMathML, png

5.12.4 $\int _{0}^{1}\frac{t^{{a-1}}(1-t)^{{b-1}}}{(t+z)^{{a+b}}}dt=\mathrm{B}\!\left(a,b\right)(1+z)^{{-a}}z^{{-b}},$ $|\mathrm{ph}z|<\pi$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E4
Encodings:: TeX, pMathML, png

5.12.5 $\int _{0}^{{\pi/2}}(\cos t)^{{a-1}}\cos\!\left(bt\right)dt=\frac{\pi}{2^{a}}\frac{1}{a\mathrm{B}\!\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E5
Encodings:: TeX, pMathML, png

5.12.6 $\int _{0}^{\pi}(\sin t)^{{a-1}}e^{{ibt}}dt=\frac{\pi}{2^{{a-1}}}\frac{e^{{i\pi b/2}}}{a\mathrm{B}\!\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E6
Encodings:: TeX, pMathML, png

5.12.7 $\int _{0}^{\infty}\frac{\cosh\!\left(2bt\right)}{(\cosh t)^{{2a}}}dt=4^{{a-1}}\mathrm{B}\!\left(a+b,a-b\right),$ $\realpart{a}>|\realpart{b}|$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E7
Encodings:: TeX, pMathML, png

5.12.8 ${\frac{1}{2\pi}\int _{{-\infty}}^{\infty}\frac{dt}{(w+it)^{a}(z-it)^{b}}=\frac{(w+z)^{{1-a-b}}}{(a+b-1)\mathrm{B}\!\left(a,b\right)}},$ $\realpart{(a+b)}>1$ , $\realpart{w}>0$ , $\realpart{z}>0$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $w$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E8
Encodings:: TeX, pMathML, png

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 ${\frac{1}{2\pi i}\int _{{c-\infty i}}^{{c+\infty i}}t^{{-a}}(1-t)^{{-1-b}}dt=\frac{1}{b\mathrm{B}\!\left(a,b\right)}},$ $0<c<1$ , $\realpart{(a+b)}>0$ .

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E9
Encodings:: TeX, pMathML, png

5.12.10 ${\frac{1}{2\pi i}\int _{0}^{{(1+)}}t^{{a-1}}(t-1)^{{b-1}}dt=\frac{\sin\!\left(\pi b\right)}{\pi}\mathrm{B}\!\left(a,b\right)},$ $\realpart{a}>0$ ,

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E10
Encodings:: TeX, pMathML, png

with the contour as shown in Figure 5.12.1.

5.12.1. $t$ -plane. Contour for first loop integral for the beta function.

Magnify

Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.F1
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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 $\frac{1}{e^{{2\pi ia}}-1}\int _{\infty}^{{(0+)}}t^{{a-1}}(1+t)^{{-a-b}}dt=\mathrm{B}\!\left(a,b\right),$

Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E11
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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.

5.12.2. $t$ -plane. Contour for second loop integral for the beta function.

Magnify

Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.F2
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¶ Pochhammer's Integral

When $a,b\in\Complex$

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

Defines:: $P$ : point
Symbols:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E12
Encodings:: TeX, pMathML, png

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.

5.12.3. $t$ -plane. Contour for Pochhammer's integral.

Magnify

Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.F3
Encodings:: eps, png