# Euler beta integral

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## 1—10 of 81 matching pages

##### 1: 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 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
##### 2: 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 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},$ $\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$.
##### 3: 35.4 Partitions and Zonal Polynomials
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: 5.12 Beta Function
###### Euler’s BetaIntegral
5.12.1 $\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}.$
5.12.4 $\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\,\mathrm{d}t=\mathrm{B}% \left(a,b\right)(1+z)^{-a}z^{-b},$ $|\operatorname{ph}z|<\pi$.
5.12.7 $\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(\cosh t)^{2a}}\,\mathrm{d}t=4^{% a-1}\mathrm{B}\left(a+b,a-b\right),$ $\Re a>|\Re b|$.
##### 5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.5 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$, $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.
##### 6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% _{1}-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{X}\right|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),$ $\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\right)>\frac{1}{2}(m-1)$.
##### 7: 19.28 Integrals of Elliptic Integrals
19.28.1 $\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)\,\mathrm{d}t=\tfrac{1}{2}\left% (\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},$
19.28.2 $\int_{0}^{1}t^{\sigma-1}R_{G}\left(0,t,1\right)\,\mathrm{d}t=\frac{\sigma}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2},$
19.28.3 $\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1\right)\,\mathrm{d}t=\frac{3}{4% \sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right)^{2}.$
##### 9: 19.33 Triaxial Ellipsoids
19.33.11 $U=\tfrac{1}{2}(\alpha\beta\gamma)^{2}R_{F}\left(\alpha^{2},\beta^{2},\gamma^{2% }\right)\int_{0}^{\infty}(g(r))^{2}\,\mathrm{d}r,$
##### 10: 16.15 Integral Representations and Integrals
16.15.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\,\mathrm{d}u,$ $\Re\alpha>0$, $\Re\left(\gamma-\alpha\right)>0$,
16.15.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\gamma>\Re\beta>0$, $\Re\gamma^{\prime}>\Re\beta^{\prime}>0$,
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.