# Euler transformation

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## 11—20 of 90 matching pages

##### 11: 13.10 Integrals
13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\mathrm{d}t=\frac% {\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a\right)% \Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
13.10.11 $\int_{0}^{\infty}t^{\lambda-1}U\left(a,b,t\right)\mathrm{d}t=\frac{\Gamma\left% (\lambda\right)\Gamma\left(a-\lambda\right)\Gamma\left(\lambda-b+1\right)}{% \Gamma\left(a\right)\Gamma\left(a-b+1\right)},$ $\max\left(\Re b-1,0\right)<\Re\lambda<\Re a$.
##### 12: 21.5 Modular Transformations
The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …
##### 13: 20.10 Integrals
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\mathrm{d}x=2^{s% }(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\mathrm{d}x=% \pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\mathrm{d}x=% (1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right).$
##### 14: 16.5 Integral Representations and Integrals
16.5.2 ${{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\mathrm{d}t,$ $\Re b_{0}>\Re a_{0}>0$,
16.5.3 ${{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\mathrm{d}t,$ $\Re z<1$, $\Re a_{0}>0$,
16.5.4 ${{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}}^{t}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\mathrm{d}t,$ $c>0$, $\Re b_{0}>0$.
##### 15: 15.6 Integral Representations
15.6.2 $\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}% \mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $c-b\neq 1,2,3,\dots$, $\Re b>0$.
15.6.3 $\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{\infty}^{(0+)}\frac{t^% {b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$.
15.6.4 $\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-b\pi\mathrm{i}}\frac{\Gamma\left% (1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}% (1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b\neq 1,2,3,\dots$, $\Re\left(c-b\right)>0$.
15.6.5 $\mathbf{F}\left(a,b;c;z\right)={\mathrm{e}}^{-c\pi\mathrm{i}}\Gamma\left(1-b% \right)\Gamma\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}% \frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $b,c-b\neq 1,2,3,\dots$.
15.6.8 $\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\mathrm{d}t,$ $|\operatorname{ph}\left(1-z\right)|<\pi$; $\Re c>\Re d>0$.
##### 16: 8.19 Generalized Exponential Integral
8.19.4 $E_{p}\left(z\right)=\frac{z^{p-1}e^{-z}}{\Gamma\left(p\right)}\int_{0}^{\infty% }\frac{t^{p-1}e^{-zt}}{1+t}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, $\Re p>0$.
##### 17: 18.17 Integrals
18.17.32 $\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax\right)\cos% \left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n)!}y^{2n}% \left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),$ $\nu>2n$, $a>0$.
18.17.36 $\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)% \mathrm{d}x=\frac{2^{\beta+z}\Gamma\left(z\right)\Gamma\left(1+\beta+n\right){% \left(1+\alpha-z\right)_{n}}}{n!\Gamma\left(1+\beta+z+n\right)},$ $\Re z>0$.
18.17.37 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)x^{z% -1}\mathrm{d}x=\frac{\pi\,2^{1-2\lambda-z}\Gamma\left(n+2\lambda\right)\Gamma% \left(z\right)}{n!\Gamma\left(\lambda\right)\Gamma\left(\frac{1}{2}+\frac{1}{2% }n+\lambda+\frac{1}{2}z\right)\Gamma\left(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}% n\right)},$ $\Re z>0$.
18.17.40 $\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\mathrm{d}x=% \frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-z}\*{{}_{2}F_{1}}\left({-n% ,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$ $\Re a>0$, $\Re z>0$.
18.17.41 $\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\Gamma% \left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+% \tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+% \tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\Re a>0$. Also, $\Re z>0$, $n$ even; $\Re z>-1$, $n$ odd.
##### 18: 10.43 Integrals
10.43.19 $\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)\mathrm{d}t=2^{\mu-2}\Gamma% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma\left(\tfrac{1}{2}\mu+\tfrac% {1}{2}\nu\right),$ $|\Re\nu|<\Re\mu$.
10.43.22 $\int_{0}^{\infty}t^{\mu-1}e^{-at}K_{\nu}\left(t\right)\mathrm{d}t=\begin{cases% }\left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)\Gamma% \left(\mu+\nu\right)(1-a^{2})^{-\frac{1}{2}\mu+\frac{1}{4}}\mathsf{P}^{-\mu+% \frac{1}{2}}_{\nu-\frac{1}{2}}\left(a\right),&-1
10.43.26 $\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{% \lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}% \lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2% }\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{\nu-\lambda+1}}\*% \mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;% -\frac{b^{2}}{a^{2}}\right),$ $\Re\left(\nu+1-\lambda\right)>|\Re\mu|,\Re a>|\Im b|$.
10.43.27 $\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at\right)J_{\nu}\left(bt\right)% \mathrm{d}t=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left(\mu+\nu+1\right)}{(a^{2}+b^{% 2})^{\mu+\nu+1}},$ $\Re\left(\nu+1\right)>|\Re\mu|,\Re a>|\Im b|$.
##### 19: 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 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.
##### 20: 16.4 Argument Unity
###### §16.4(iii) Identities
Balanced ${{}_{4}F_{3}}\left(1\right)$ series have transformation formulas and three-term relations. The basic transformation is given by … A different type of transformation is that of Whipple: … See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …