# §5.8 Infinite Products

 5.8.1 $\mathop{\Gamma\/}\nolimits\!\left(z\right)=\lim_{k\to\infty}\frac{k!k^{z}}{z(z% +1)\cdots(z+k)},$ $z\neq 0,-1,-2,\dots$, Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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
 5.8.2 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(z\right)}=ze^{\gamma z}\prod_{k=1}^% {\infty}\left(1+\frac{z}{k}\right)e^{-z/k},$
 5.8.3 $\left|\frac{\mathop{\Gamma\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}% \nolimits\!\left(x+\mathrm{i}y\right)}\right|^{2}=\prod_{k=0}^{\infty}\left(1+% \frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$. Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $k$: 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

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

then

 5.8.5 $\prod_{k=0}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k% )\cdots(b_{m}+k)}=\frac{\mathop{\Gamma\/}\nolimits\!\left(b_{1}\right)\mathop{% \Gamma\/}\nolimits\!\left(b_{2}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(% b_{m}\right)}{\mathop{\Gamma\/}\nolimits\!\left(a_{1}\right)\mathop{\Gamma\/}% \nolimits\!\left(a_{2}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(a_{m}% \right)},$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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

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