5 Gamma Function Properties 5.11 Asymptotic Expansions 5.13 Integrals

§5.12 Beta Function

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Keywords:: beta function
Notes:: For (5.12.1)–(5.12.4) see Carlson (1977b, pp. 60 and 70). For (5.12.5)–(5.12.6) and (5.12.8) see Nielsen (1906a, §64). For (5.12.7), (5.12.9), (5.12.10), and (5.12.12) see Temme (1996b, pp. 74–75) and Olver (1997b, p. 38). (An error in Ex.3.13 of Temme (1996b) is corrected here.) (5.12.11) follows from (5.12.3).
Referenced by:: §18.15(i), §18.17(v), §19.1, §19.16(ii), §19.23, §19.28, §8.17(i)
Permalink:: http://dlmf.nist.gov/5.12
See also:: Annotations for Ch.5

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

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Keywords:: Euler’s beta integral, beta function, beta integrals, definition, integral representations
See also:: Annotations for §5.12 and Ch.5

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 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 See also: Annotations for §5.12, §5.12 and Ch.5

5.12.2		$\int_{0}^{\pi/2}{\sin}^{2a-1}\theta{\cos}^{2b-1}\theta\,\mathrm{d}\theta=% \tfrac{1}{2}\mathrm{B}\left(a,b\right).$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : 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) Permalink: http://dlmf.nist.gov/5.12.E2 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

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 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: §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		$\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t z)^{a b}}\,\mathrm{d}t=\mathrm{B}% \left(a,b\right)(1 z)^{-a}z^{-b},$
$\|\operatorname{ph}z\|<\pi$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : 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, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

5.12.5		$\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)\,\mathrm{d}t=\frac{\pi}{2^{a% }}\frac{1}{a\mathrm{B}\left(\frac{1}{2}(a b 1),\frac{1}{2}(a-b 1)\right)},$
$\Re a>0$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §10.22(ii), §5.12, §7.6(ii) Permalink: http://dlmf.nist.gov/5.12.E5 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

5.12.6		$\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\,\mathrm{d}t=\frac{\pi}{2^{a-1}}\frac{e^{i% \pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a b 1),\frac{1}{2}(a-b 1)\right)},$
$\Re a>0$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §10.22(ii), §5.12 Permalink: http://dlmf.nist.gov/5.12.E6 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

5.12.7		$\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(\cosh t)^{2a}}\,\mathrm{d}t=4^{% a-1}\mathrm{B}\left(a b,a-b\right),$
$\Re a>\|\Re b\|$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.12 Permalink: http://dlmf.nist.gov/5.12.E7 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

5.12.8		${\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\,\mathrm{d}t}{(w it)^{a}(z-it)^{b% }}=\frac{(w z)^{1-a-b}}{(a b-1)\mathrm{B}\left(a,b\right)}},$
$\Re\left(a b\right)>1$ , $\Re w>0$ , $\Re z>0$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $w$ : 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, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

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		${\frac{1}{2\pi i}\int_{c-\infty i}^{c \infty i}t^{-a}(1-t)^{-1-b}\,\mathrm{d}t% =\frac{1}{b\mathrm{B}\left(a,b\right)}},$
$0<c<1$ , $\Re\left(a b\right)>0$ .
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.12 Permalink: http://dlmf.nist.gov/5.12.E9 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

5.12.10		${\frac{1}{2\pi i}\int_{0}^{(1 )}t^{a-1}(t-1)^{b-1}\,\mathrm{d}t=\frac{\sin% \left(\pi b\right)}{\pi}\mathrm{B}\left(a,b\right)},$
$\Re a>0$ ,
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine 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, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

with the contour as shown in Figure 5.12.1.

See accompanying text — 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		$\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0 )}t^{a-1}(1 t)^{-a-b}\,\mathrm{d}t=% \mathrm{B}\left(a,b\right),$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $a$ : real or complex variable and $b$ : real or complex variable Referenced by: §5.12, §5.12 Permalink: http://dlmf.nist.gov/5.12.E11 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

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.

Pochhammer’s Integral

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Keywords:: Pochhammer’s integral, beta function
See also:: Annotations for §5.12 and Ch.5

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

5.12.12		$\int_{P}^{(1 ,0 ,1-,0-)}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t=-4e^{\pi i(a b)}\sin% \left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}\left(a,b\right),$
ⓘ Symbols: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex variable, $b$ : real or complex variable and $P$ : point Referenced by: §5.12, §5.12 Permalink: http://dlmf.nist.gov/5.12.E12 Encodings: TeX, pMML, png See also: Annotations for §5.12, §5.12 and Ch.5

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.

5.11 Asymptotic Expansions 5.13 Integrals

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