# Euler integral

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## 1—10 of 206 matching pages

##### 1: 17.13 Integrals
17.13.2 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.$
17.13.3 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},$
17.13.4 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
##### 2: 5.2 Definitions
###### Euler’s Integral
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\,\mathrm{d}t,$ $\Re z>0$.
##### 3: 6.11 Relations to Other Functions
6.11.1 $E_{1}\left(z\right)=\Gamma\left(0,z\right).$
##### 4: 35.3 Multivariate Gamma and Beta Functions
35.3.1 $\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-% \mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$.
35.3.2 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}% \operatorname{etr}\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac% {1}{2}(m+1)}\prod_{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\,\mathrm{d}{% \mathbf{X}},$ $s_{j}\in\mathbb{C}$, $\Re\left(s_{j}\right)>\frac{1}{2}(j-1)$, $j=1,\dots,m$.
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 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
##### 5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 $\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.6.4 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+% 1)}\left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)$.
35.6.5 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};% \mathbf{S}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)% \left|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}\right|^{-a}\left|\mathbf{T}\right|^% {-b},$ $\mathbf{T}>\mathbf{S}$, $\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.6.6 $\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},$ $\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)$.
35.6.8 $\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\,\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},$ $\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\Re\left(c-b\right)>-1$.
##### 6: 24.13 Integrals
###### §24.13(ii) Euler Polynomials
24.13.8 $\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
24.13.10 $\int_{0}^{1/2}E_{2n-1}\left(t\right)\,\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}},$ $n=1,2,\dots$.
##### 7: 8.22 Mathematical Applications
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
8.22.2 $\zeta_{x}(s)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{x}\frac{t^{s-1}}{e^{t}-1}% \,\mathrm{d}t,$ $\Re s>1$,
##### 8: 24.7 Integral Representations
###### §24.7(i) Bernoulli and Euler Numbers
24.7.5 $B_{2n}=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\ln\left(1-e^{-2% \pi t}\right)\,\mathrm{d}t.$
24.7.6 $E_{2n}=(-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\operatorname{sech}\left(\pi t% \right)\,\mathrm{d}t.$
###### §24.7(ii) Bernoulli and Euler Polynomials
24.7.9 $E_{2n}\left(x\right)=(-1)^{n}4\int_{0}^{\infty}\frac{\sin\left(\pi x\right)% \cosh\left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2% n}\,\mathrm{d}t,$
##### 10: 8.4 Special Values
8.4.1 $\gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{-t^{2}}\,\mathrm{d}t=% \sqrt{\pi}\operatorname{erf}\left(z\right),$
8.4.4 $\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}\,\mathrm{d}t=E_{1}\left(z% \right),$
8.4.6 $\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{d}% t=\sqrt{\pi}\operatorname{erfc}\left(z\right).$
8.4.15 $\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}% \sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=\frac{(-1)^{n}}{n!}\left(% \psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.$