# §5.5 Functional Relations

## §5.5(i) Recurrence

 5.5.1 $\Gamma\left(z+1\right)=z\Gamma\left(z\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $z$: complex variable A&S Ref: 6.1.15 Referenced by: §15.4(ii), §15.4(iii), (25.11.27), (25.5.6), §5.4(i) Permalink: http://dlmf.nist.gov/5.5.E1 Encodings: TeX, pMML, png See also: Annotations for §5.5(i), §5.5 and Ch.5
 5.5.2 $\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}.$ ⓘ Symbols: $\psi\left(\NVar{z}\right)$: psi (or digamma) function and $z$: complex variable A&S Ref: 6.3.5 Permalink: http://dlmf.nist.gov/5.5.E2 Encodings: TeX, pMML, png See also: Annotations for §5.5(i), §5.5 and Ch.5

## §5.5(ii) Reflection

 5.5.3 $\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 6.1.17 (without the condition on $z$.) Referenced by: §10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15) Permalink: http://dlmf.nist.gov/5.5.E3 Encodings: TeX, pMML, png See also: Annotations for §5.5(ii), §5.5 and Ch.5
 5.5.4 $\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\tan\NVar{z}$: tangent function and $z$: complex variable A&S Ref: 6.3.7 (without the condition on $z$.) Permalink: http://dlmf.nist.gov/5.5.E4 Encodings: TeX, pMML, png See also: Annotations for §5.5(ii), §5.5 and Ch.5

## §5.5(iii) Multiplication

### Duplication Formula

For $2z\neq 0,-1,-2,\dots$,

 5.5.5 $\Gamma\left(2z\right)=\pi^{-1/2}2^{2z-1}\Gamma\left(z\right)\Gamma\left(z+% \tfrac{1}{2}\right).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable A&S Ref: 6.1.18 Referenced by: §14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E5 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

### Gauss’s Multiplication Formula

For $nz\neq 0,-1,-2,\dots$,

 5.5.6 $\Gamma\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left% (z+\frac{k}{n}\right).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable A&S Ref: 6.1.20 Referenced by: §15.4(iii), §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E6 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5
 5.5.7 $\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}.$
 5.5.8 $\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}% {2}\right)\right)+\ln 2,$ ⓘ Symbols: $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 6.3.8 Referenced by: §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E8 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5
 5.5.9 $\psi\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}\psi\left(z+\frac{k}{n}\right)+% \ln n.$

If a positive function $f(x)$ on $(0,\infty)$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\Gamma\left(x\right)$.