# Pochhammer integral

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## 1—10 of 51 matching pages

##### 1: 7.12 Asymptotic Expansions
7.12.2 $\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
7.12.3 $\mathrm{g}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},$
7.12.4 $\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\mathrm{f})}(z),$
7.12.5 $\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\mathrm{g})}(z),$
##### 2: 17.13 Integrals
17.13.1 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}\,{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},$
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 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
##### 3: 5.12 Beta Function
###### Pochhammer’s Integral
When $a,b\in\mathbb{C}$
##### 4: 31.9 Orthogonality
31.9.2 $\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.$
18.28.8 $\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;a,b\,|\,q\right)Q_{m}\left(% \cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left({\mathrm{e}}^{2i\theta};q% \right)_{\infty}}{\left(a{\mathrm{e}}^{\mathrm{i}\theta},b{\mathrm{e}}^{% \mathrm{i}\theta};q\right)_{\infty}}\right|}^{2}\,\mathrm{d}\theta=\frac{% \delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},$ $a,b\in\mathbb{R}$ or $a=\overline{b}$; $ab\neq 1$; $|a|,|b|\leq 1$.
18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,|\,q\right)C_{m}\left% (\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left({\mathrm{e}}^{2\mathrm{i}% \theta};q\right)_{\infty}}{\left(\beta{\mathrm{e}}^{2\mathrm{i}\theta};q\right% )_{\infty}}\right|}^{2}\,\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{% \infty}}{\left(\beta^{2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q% \right)_{n}}{(1-\beta q^{n})\left(q;q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.
18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta\,|\,q\right)H_{m}\left(\cos% \theta\,|\,q\right){\left|\left({\mathrm{e}}^{2\mathrm{i}\theta};q\right)_{% \infty}\right|}^{2}\,\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q% \right)_{\infty}}.$
##### 6: 19.5 Maclaurin and Related Expansions
19.5.1 $K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
19.5.3 $D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{% }_{2}F_{1}}\left({\tfrac{3}{2},\tfrac{1}{2}\atop 2};k^{2}\right),$
19.5.4_1 $F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),$
19.5.4_2 $E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),$
19.5.4_3 $\Pi\left(\phi,\alpha^{2},k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\sin}^{2m+1}\phi}{(2m+1)m!}{F_{1}}\left(m+\tfrac{1}{2};\tfrac{1}{% 2},1;m+\tfrac{3}{2};{\sin}^{2}\phi,\alpha^{2}{\sin}^{2}\phi\right)k^{2m},$
##### 7: 19.12 Asymptotic Approximations
19.12.1 $K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),$ $0<|k^{\prime}|<1$,
19.12.2 $E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),$ $|k^{\prime}|<1$,
##### 8: 25.5 Integral Representations
25.5.7 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\,\mathrm{d}x,$ $\Re s>-(2n+1)$, $n=1,2,3,\dots$.
##### 9: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$
8.20.1 $E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p% }\left(z\right)\right),$ $n=1,2,3,\dots$.
8.20.2 $E_{p}\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}},$ $|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta$,
8.20.3 $E_{p}\left(z\right)\sim\pm\frac{2\pi i}{\Gamma\left(p\right)}e^{\mp p\pi i}z^{% p-1}+\frac{e^{-z}}{z}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(p\right)_{k}}}{z^% {k}},$ $\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{7}{2}\pi-\delta$,
##### 10: 8.21 Generalized Sine and Cosine Integrals
8.21.16 $\operatorname{Si}\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac% {3}{2}\right){\left(1-\frac{1}{2}a\right)_{k}}}{{\left(\frac{1}{2}+\frac{1}{2}% a\right)_{k+1}}}\mathsf{j}_{2k+1}\left(z\right),$ $a\neq-1,-3,-5,\dots$,
8.21.17 $\operatorname{Ci}\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac% {1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)_{k}}}{{\left(\frac{1}{2}a% \right)_{k+1}}}\mathsf{j}_{2k}\left(z\right),$ $a\neq 0,-2,-4,\dots$.