5 Gamma Function Properties 5.7 Series Expansions 5.9 Integral Representations

§5.8 Infinite Products

ⓘ

Keywords:: gamma function, infinite products
Notes:: For (5.8.1)–(5.8.3) see Olver (1997b, pp. 34 and 38). For (5.8.4)–(5.8.5) see Whittaker and Watson (1927, p. 238–239).
Permalink:: http://dlmf.nist.gov/5.8
See also:: Annotations for Ch.5

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 ), : nonnegative integer and : 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, : nonnegative integer and : 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, : nonnegative integer, : real variable and : 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

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:: : nonnegative integer and : 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, : nonnegative integer, : 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.

5.7 Series Expansions 5.9 Integral Representations

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov