# beta function

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##### 1: 5.12 Beta Function
###### Euler’s Beta Integral Figure 5.12.1: t -plane. Contour for first loop integral for the beta function. Magnify Figure 5.12.2: t -plane. Contour for second loop integral for the beta function. Magnify
##### 3: 8.23 Statistical Applications
###### §8.23 Statistical Applications
The function $\mathrm{B}_{x}\left(a,b\right)$ and its normalization $I_{x}\left(a,b\right)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
##### 5: 5.20 Physical Applications
###### §5.20 Physical Applications
Then the partition function (with $\beta=1/(kT)$) is given by …
###### Elementary Particles
Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …
##### 7: 5.18 $q$-Gamma and $q$-Beta Functions
###### §5.18(iii) $q$-BetaFunction
5.18.11 $\mathrm{B}_{q}\left(a,b\right)=\frac{\Gamma_{q}\left(a\right)\Gamma_{q}\left(b% \right)}{\Gamma_{q}\left(a+b\right)}.$
5.18.12 $\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{% \infty}}{\left(tq^{b};q\right)_{\infty}}{\mathrm{d}}_{q}t,$ $0, $\Re a>0$, $\Re b>0$.
##### 8: 35.3 Multivariate Gamma and Beta Functions
###### §35.3 Multivariate Gamma and BetaFunctions
35.3.3 ${}\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.3.7 $\mathrm{B}_{m}\left(a,b\right)=\frac{\Gamma_{m}\left(a\right)\Gamma_{m}\left(b% \right)}{\Gamma_{m}\left(a+b\right)}.$
35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\mathrm% {d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
##### 9: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$
8.18.1 $I_{x}\left(a,b\right)={\Gamma\left(a+b\right)x^{a}(1-x)^{b-1}}\*\left(\sum_{k=% 0}^{n-1}\frac{1}{\Gamma\left(a+k+1\right)\Gamma\left(b-k\right)}\left(\frac{x}% {1-x}\right)^{k}+O\left(\frac{1}{\Gamma\left(a+n+1\right)}\right)\right),$
##### 10: 8.1 Special Notation
Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions $\gamma\left(a,z\right)$, $\Gamma\left(a,z\right)$, $\gamma^{*}\left(a,z\right)$, $P\left(a,z\right)$, and $Q\left(a,z\right)$; the incomplete beta functions $\mathrm{B}_{x}\left(a,b\right)$ and $I_{x}\left(a,b\right)$; the generalized exponential integral $E_{p}\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}\left(a,z\right)$, $\mathrm{ci}\left(a,z\right)$, $\mathrm{Si}\left(a,z\right)$, and $\mathrm{Ci}\left(a,z\right)$. Alternative notations include: Prym’s functions $P_{z}(a)=\gamma\left(a,z\right)$, $Q_{z}(a)=\Gamma\left(a,z\right)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma\left(a+1,z\right)$, $[a,z]!=\Gamma\left(a+1,z\right)$, Dingle (1973); $B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)$, $I(a,b,x)=I_{x}\left(a,b\right)$, Magnus et al. (1966); $\mathrm{Si}\left(a,x\right)\to\mathrm{Si}\left(1-a,x\right)$, $\mathrm{Ci}\left(a,x\right)\to\mathrm{Ci}\left(1-a,x\right)$, Luke (1975).