5 Gamma Function Properties 5.11 Asymptotic Expansions 5.13 Integrals

§5.12 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).
Keywords:: beta function
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

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

Keywords:: Euler’s beta integral, beta function, beta integrals

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, : differential of , $\int$ : integral, : real or complex variable and : 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 $\int_{0}^{{\pi/2}}{\mathop{\sin\/}\nolimits^{{2a-1}}}\theta{\mathop{\cos\/}% \nolimits^{{2b-1}}}\theta d\theta=\tfrac{1}{2}\mathop{\mathrm{B}\/}\nolimits\!% \left(a,b\right).$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex variable and : 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

5.12.3 $\int_{0}^{\infty}\frac{t^{{a-1}}dt}{(1+t)^{{a+b}}}=\mathop{\mathrm{B}\/}% \nolimits\!\left(a,b\right).$

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral, : real or complex variable and : 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

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable, : real or complex variable and : real or complex variable
Referenced by:: §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E4
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real or complex variable and : 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

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex variable and : real or complex variable
Referenced by:: §10.22(ii), §5.12
Permalink:: http://dlmf.nist.gov/5.12.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\realpart{}$ : real part, : real or complex variable and : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E7
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real or complex variable, : complex variable, : real or complex variable and : real or complex variable
Referenced by:: ¶ ‣ §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E8
Encodings:: TeX, pMML, png

In (5.12.8) the fractional powers have their principal values when and , and are continued via continuity.

5.12.9 ${\frac{1}{2\pi i}\int_{{c-\infty i}}^{{c+\infty i}}t^{{-a}}(1-t)^{{-1-b}}dt=% \frac{1}{b\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)}},$

, $\realpart{(a+b)}>0$ .

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral, $\realpart{}$ : real part, : real or complex variable and : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E9
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex variable and : real or complex variable
Referenced by:: §5.12
Permalink:: http://dlmf.nist.gov/5.12.E10
Encodings:: TeX, pMML, png

with the contour as shown in Figure 5.12.1.

Figure 5.12.1:

-plane. Contour for first loop integral for the beta function.

Referenced by:: ¶ ‣ §5.12
Permalink:: http://dlmf.nist.gov/5.12.F1
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , : base of exponential function, $\int$ : integral, : real or complex variable and : real or complex variable
Referenced by:: ¶ ‣ §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E11
Encodings:: TeX, pMML, png

when $\realpart{b}>0$ , 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:

-plane. Contour for second loop integral for the beta function.

Referenced by:: ¶ ‣ §5.12
Permalink:: http://dlmf.nist.gov/5.12.F2
Encodings:: pdf, png

¶ Pochhammer’s Integral

Keywords:: Pochhammer’s integral, beta function

When $a,b\in\Complex$

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

Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex variable, : real or complex variable and : point
Referenced by:: ¶ ‣ §5.12, §5.12
Permalink:: http://dlmf.nist.gov/5.12.E12
Encodings:: TeX, pMML, png

where the contour starts from an arbitrary point in the interval , circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to . It can always be deformed into the contour shown in Figure 5.12.3.

Figure 5.12.3:

-plane. Contour for Pochhammer’s integral.

Referenced by:: Figure 13.4.1, Figure 13.4.1, §15.6, §31.9(i), ¶ ‣ §5.12
Permalink:: http://dlmf.nist.gov/5.12.F3
Encodings:: pdf, png

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