# beta integrals

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## 11—20 of 98 matching pages

##### 11: Bibliography P
• P. I. Pastro (1985) Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
##### 13: 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).
##### 14: 19.26 Addition Theorems
19.26.13 $R_{C}\left(\alpha^{2},\alpha^{2}-\theta\right)+R_{C}\left(\beta^{2},\beta^{2}-% \theta\right)=R_{C}\left(\sigma^{2},\sigma^{2}-\theta\right),$ $\sigma=(\alpha\beta+\theta)/(\alpha+\beta)$,
19.26.17 $\sqrt{\alpha}R_{C}\left(\beta,\alpha+\beta\right)+\sqrt{\beta}R_{C}\left(% \alpha,\alpha+\beta\right)=\pi/2,$ $\alpha,\beta\in\mathbb{C}\setminus(-\infty,0)$, $\alpha+\beta>0$.
19.26.22 $R_{J}\left(x,y,z,p\right)=2R_{J}\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda% \right)+3R_{C}\left(\alpha^{2},\beta^{2}\right),$
##### 15: 8.17 Incomplete Beta Functions
8.17.1 $\mathrm{B}_{x}\left(a,b\right)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}\mathrm{d}t,$
###### §8.17(iii) Integral Representation
8.17.10 $I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{2\pi i}\int_{c-i\infty}^{c+i\infty% }s^{-a}(1-s)^{-b}\frac{\mathrm{d}s}{s-x},$
##### 16: Bibliography
• W. A. Al-Salam and M. E. H. Ismail (1994) A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
• ##### 17: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.5 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$, $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.
##### 18: 5.20 Physical Applications
5.20.3 $\psi_{n}(\beta)=\int_{{\mathbb{R}}^{n}}e^{-\beta W}\mathrm{d}x\\ =(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}% \beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right).$
5.20.5 $\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}e^{-\beta W}\mathrm{d% }\theta_{1}\cdots\mathrm{d}\theta_{n}=\Gamma\left(1+\tfrac{1}{2}n\beta\right)(% \Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}.$
##### 19: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% _{1}-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{X}\right|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),$ $\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\right)>\frac{1}{2}(m-1)$.
##### 20: 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 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.