# §5.8 Infinite Products

 5.8.1 $\Gamma\left(z\right)=\lim_{k\to\infty}\frac{k!k^{z}}{z(z+1)\cdots(z+k)},$ $z\neq 0,-1,-2,\dots$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.1.2 Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E1 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5
 5.8.2 $\frac{1}{\Gamma\left(z\right)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{% z}{k}\right)e^{-z/k},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.1.3 Permalink: http://dlmf.nist.gov/5.8.E2 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5
 5.8.3 $\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+\mathrm{i}y\right)}\right|^{2}% =\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable A&S Ref: 6.1.25 (where the formula for the reciprocal is given.) Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E3 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

If

 5.8.4 $\sum_{k=1}^{m}a_{k}=\sum_{k=1}^{m}b_{k},$ ⓘ Defines: $a_{k}$: coefficient (locally) and $b_{k}$: coefficient (locally) Symbols: $m$: nonnegative integer and $k$: nonnegative integer Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E4 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

then

 5.8.5 $\prod_{k=0}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k% )\cdots(b_{m}+k)}=\frac{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\cdots% \Gamma\left(b_{m}\right)}{\Gamma\left(a_{1}\right)\Gamma\left(a_{2}\right)% \cdots\Gamma\left(a_{m}\right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_{k}$: coefficient and $b_{k}$: coefficient Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E5 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

provided that none of the $b_{k}$ is zero or a negative integer.