DLMF: GA.12 Beta Function

§GA.12. Beta Function

Sources:

Carlson(1977)(§4.2 and p. 70), Nielsen(1906)(§64), Temme(1996)(§3.8: an error in Ex.3.13 is corrected), Olver(1997)(p. 38). (GA.12.11) follows from (GA.12.3).

Notes:

GA.12.1

GA.12.4

Carlson(1977)(pp. 60 and 70)

Nielsen(1906)(§64)

Temme(1996)(pp. 74–75)

Olver(1997)(p. 38)

Keywords:

In this section all fractional powers have their principal values, except where noted otherwise. In (GA.12.1)–(GA.12.4) it is assumed $\realpart{a}>0$ and $\realpart{b}>0$ .

Euler's Beta Integral

Keywords:: beta integral, Euler's beta integral

GA.12.1 $\mathrm{B}\!\left(a,b\right)=\int _{0}^{1}t^{{a-1}}(1-t)^{{b-1}}dt=\frac{\Gamma\!\left(a\right)\Gamma\!\left(b\right)}{\Gamma\!\left(a+b\right)}.$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $\Gamma\!\left(z\right)$ : 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:: §GA.12, §GA.12
Encodings:: pMathML, png, TeX

GA.12.2 $\int _{0}^{{\pi/2}}{\mathrm{sin}^{{2a-1}}}\theta{\mathrm{cos}^{{2b-1}}}\theta d\theta=\tfrac{1}{2}\mathrm{B}\!\left(a,b\right),$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
A&S Ref:: 6.2.1 and 6.2.2
Encodings:: pMathML, png, TeX

GA.12.3 $\int _{0}^{\infty}\frac{t^{{a-1}}dt}{(1+t)^{{a+b}}}=\mathrm{B}\!\left(a,b\right),$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : 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:: §GA.12
Encodings:: pMathML, png, TeX

GA.12.4 $\int _{0}^{1}\frac{t^{{a-1}}(1-t)^{{b-1}}}{(t+z)^{{a+b}}}dt=\mathrm{B}\!\left(a,b\right)(1+z)^{{-a}}z^{{-b}},$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12, §GA.12
Encodings:: pMathML, png, TeX

with $|\mathrm{ph}z|<\pi$ and the integration path along the real axis.

GA.12.5 $\int _{0}^{{\pi/2}}(\mathrm{cos}t)^{{a-1}}\mathrm{cos}\!\left(bt\right)dt=\frac{\pi}{2^{a}}\frac{1}{a\mathrm{B}\!\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$ ,

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

GA.12.6 $\int _{0}^{\pi}(\mathrm{sin}t)^{{a-1}}e^{{ibt}}dt=\frac{\pi}{2^{{a-1}}}\frac{e^{{i\pi b/2}}}{a\mathrm{B}\!\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},$ $\realpart{a}>0$ ,

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

GA.12.7 $\int _{0}^{\infty}\frac{\mathrm{cosh}\!\left(2bt\right)}{(\mathrm{cosh}t)^{{2a}}}dt=4^{{a-1}}\mathrm{B}\!\left(a+b,a-b\right),$ $\realpart{a}>|\realpart{b}|$ .

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

GA.12.8 ${\frac{1}{2\pi}\int _{{-\infty}}^{\infty}\frac{dt}{(w+it)^{a}(z-it)^{b}}=\frac{(w+z)^{{1-a-b}}}{(a+b-1)\mathrm{B}\!\left(a,b\right)}},$ $\realpart{(a+b)}>1$ , $\realpart{w}>0$ , $\realpart{z}>0$ .

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $w$ : real or complex variable, $z$ : complex variable, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

The fractional powers have their principal values when $w>0$ and $z>0$ , and are continued via continuity.

GA.12.9 ${\frac{1}{2\pi i}\int _{{c-\infty i}}^{{c+\infty i}}t^{{-a}}(1-t)^{{-1-b}}dt=\frac{1}{b\mathrm{B}\!\left(a,b\right)}},$ $0<c<1$ , $\realpart{(a+b)}>0$ .

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

GA.12.10 ${\frac{1}{2\pi i}\int _{0}^{{(1+)}}t^{{a-1}}(t-1)^{{b-1}}dt=\frac{\mathrm{sin}\!\left(\pi b\right)}{\pi}\mathrm{B}\!\left(a,b\right)},$ $\realpart{a}>0$ ,

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12
Encodings:: pMathML, png, TeX

with the contour as shown in Figure GA.12.1.

GA.12.1 $t$ -plane. Contour for first loop integral for the beta function.

Magnify

Referenced by:: §GA.12
Encodings:: eps, png

In (GA.12.11) and (GA.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning.

GA.12.11 $\frac{1}{e^{{2\pi ia}}-1}\int _{\infty}^{{(0+)}}t^{{a-1}}(1+t)^{{-a-b}}dt=\mathrm{B}\!\left(a,b\right),$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable and $b$ : real or complex variable
Referenced by:: §GA.12, §GA.12
Encodings:: pMathML, png, TeX

when $\realpart{b}>0$ , $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. See Figure GA.12.2.

GA.12.2 $t$ -plane. Contour for second loop integral for the beta function.

Magnify

Referenced by:: §GA.12
Encodings:: eps, png

Pochhammer's Integral

Keywords:: beta function, beta integral, Pochhammer's integral

When $a,b\in\Complex$

GA.12.12 $\int _{P}^{{(1+,0+,1-,0-)}}t^{{a-1}}(1-t)^{{b-1}}dt=-4e^{{\pi i(a+b)}}\mathrm{sin}\!\left(\pi a\right)\mathrm{sin}\!\left(\pi b\right)\mathrm{B}\!\left(a,b\right),$

Notations:: $\mathrm{B}\!\left(a,b\right)$ : Beta function, $a$ : real or complex variable, $b$ : real or complex variable and $P$ : point
Referenced by:: §GA.12, §GA.12
Encodings:: pMathML, png, TeX

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 GA.12.3.

GA.12.3 $t$ -plane. Contour for Pochhammer's integral.

Magnify

Referenced by:: §GA.12
Encodings:: eps, png