beta integrals

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2: 5.12 Beta Function
Euler’s BetaIntegral
5.12.1 $\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}.$
3: 35.4 Partitions and Zonal Polynomials
Laplace and BetaIntegrals
35.4.9 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% -\frac{1}{2}(m+1)}\*\left|\mathbf{I}-\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}Z_{% \kappa}\left(\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{{\left[% a\right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathrm{B}_{m}\left(a,b\right)% Z_{\kappa}\left(\mathbf{T}\right).$
4: 19.7 Connection Formulas
$F\left(\phi,k_{1}\right)=kF\left(\beta,k\right),$
$E\left(\phi,k_{1}\right)=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k% \right))/k,$
$\Pi\left(\phi,\alpha^{2},k_{1}\right)=k\Pi\left(\beta,k^{2}\alpha^{2},k\right),$ $k_{1}=1/k$, $\sin\beta=k_{1}\sin\phi\leq 1$.
5: Bibliography I
• M. E. H. Ismail and D. R. Masson (1994) $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
• 6: 29.18 Mathematical Applications
$\beta=K+\mathrm{i}\beta^{\prime},$
$0\leq\beta^{\prime}\leq 2{K^{\prime}},$
$\beta=K+\mathrm{i}\beta^{\prime},$ $0\leq\beta^{\prime}\leq 2{K^{\prime}},0\leq\gamma\leq 4K$,
8: 35.3 Multivariate Gamma and Beta Functions
35.3.3 $\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
9: 5.18 $q$-Gamma and $q$-Beta Functions
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, $\Re a>0$, $\Re b>0$.
10: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+% 1)}\left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$.
35.6.6 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},$ $\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$.