# Euler integral

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## 21—30 of 199 matching pages

##### 21: 8.8 Recurrence Relations and Derivatives
8.8.14 $\left.\frac{\partial}{\partial a}\gamma^{*}\left(a,z\right)\right|_{a=0}=-E_{1% }\left(z\right)-\ln z.$
##### 22: 5.9 Integral Representations
5.9.3 $c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\mathrm% {d}t,$ $c>0$, $\Re z>0$,
5.9.4 $\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t}\mathrm{d}t+\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{(z+k)k!},$ $z\neq 0,-1,-2,\dots$.
##### 23: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.5 ${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$, $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$.
##### 24: 6.10 Other Series Expansions
6.10.6 $\mathrm{Ei}\left(x\right)=\gamma+\ln\left|x\right|+\sum_{n=0}^{\infty}(-1)^{n}% (x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{2},$ $x\neq 0$,
##### 25: 8.19 Generalized Exponential Integral
8.19.1 $E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z\right).$
Most properties of $E_{p}\left(z\right)$ follow straightforwardly from those of $\Gamma\left(a,z\right)$. …
8.19.4 $E_{p}\left(z\right)=\frac{z^{p-1}e^{-z}}{\Gamma\left(p\right)}\int_{0}^{\infty% }\frac{t^{p-1}e^{-zt}}{1+t}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, $\Re p>0$.
8.19.10 $E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p\right)-\sum_{k=0}^{\infty}\frac{(-z% )^{k}}{k!(1-p+k)},$
8.19.11 $E_{p}\left(z\right)=\Gamma\left(1-p\right)\left(z^{p-1}-e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{\Gamma\left(2-p+k\right)}\right),$
##### 26: 6.4 Analytic Continuation
6.4.3 $E_{1}\left(ze^{\pm\pi i}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma\mp\pi i,$ $|\operatorname{ph}z|\leq\pi$.
##### 27: 25.12 Polylogarithms
25.12.11 $\mathrm{Li}_{s}\left(z\right)\equiv\frac{z}{\Gamma\left(s\right)}\int_{0}^{% \infty}\frac{x^{s-1}}{e^{x}-z}\mathrm{d}x,$
25.12.14 $F_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}% +1}\mathrm{d}t,$ $s>-1$,
25.12.15 $G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}% -1}\mathrm{d}t,$ $s>-1$, $x<0$; or $s>0$, $x\leq 0$,
##### 28: 29.18 Mathematical Applications
$0\leq\gamma\leq 4K,$
$\beta=K+\mathrm{i}\beta^{\prime},$ $0\leq\beta^{\prime}\leq 2{K^{\prime}},0\leq\gamma\leq 4K$,
##### 30: 13.4 Integral Representations
13.4.3 ${\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\mathrm{d}t,$ $\Re a>0$.
13.4.4 $U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-zt}t^{a% -1}(1+t)^{b-a-1}\mathrm{d}t,$ $\Re a>0$, $|\operatorname{ph}{z}|<\frac{1}{2}\pi$,
13.4.9 ${\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1+a-b\right)}{2\pi\mathrm{i}% \Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\mathrm{d}t,$ $b-a\neq 1,2,3,\dots$, $\Re a>0$.
13.4.16 ${\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(-t\right)}{\Gamma\left(b+t\right)}z^{t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,