§5.5 Functional Relations

§5.5(i) Recurrence

 5.5.1 $\mathop{\Gamma\/}\nolimits\!\left(z+1\right)=z\mathop{\Gamma\/}\nolimits\!% \left(z\right),$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function and $z$: complex variable A&S Ref: 6.1.15 Referenced by: §15.4(ii), §15.4(iii), §25.5(i), §5.4(i) Permalink: http://dlmf.nist.gov/5.5.E1 Encodings: TeX, pMML, png See also: Annotations for 5.5(i)
 5.5.2 $\mathop{\psi\/}\nolimits\!\left(z+1\right)=\mathop{\psi\/}\nolimits\!\left(z% \right)+\frac{1}{z}.$ Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: 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(ii) Reflection

 5.5.3 $\mathop{\Gamma\/}\nolimits\!\left(z\right)\mathop{\Gamma\/}\nolimits\!\left(1-% z\right)=\pi/\mathop{\sin\/}\nolimits\!\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$, Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sin\/}\nolimits\NVar{z}$: 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), §25.11(vii), §5.21, 9.10.18, §9.12(vi) Permalink: http://dlmf.nist.gov/5.5.E3 Encodings: TeX, pMML, png See also: Annotations for 5.5(ii)
 5.5.4 $\mathop{\psi\/}\nolimits\!\left(z\right)-\mathop{\psi\/}\nolimits\!\left(1-z% \right)=-\pi/\mathop{\tan\/}\nolimits\!\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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(iii) Multiplication

Duplication Formula

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

 5.5.5 $\mathop{\Gamma\/}\nolimits\!\left(2z\right)=\pi^{-1/2}2^{2z-1}\mathop{\Gamma\/% }\nolimits\!\left(z\right)\mathop{\Gamma\/}\nolimits\!\left(z+\tfrac{1}{2}% \right).$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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), §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E5 Encodings: TeX, pMML, png See also: Annotations for 5.5(iii)

Gauss’s Multiplication Formula

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

 5.5.6 $\mathop{\Gamma\/}\nolimits\!\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_% {k=0}^{n-1}\mathop{\Gamma\/}\nolimits\!\left(z+\frac{k}{n}\right).$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 5.5.7 $\prod_{k=1}^{n-1}\mathop{\Gamma\/}\nolimits\!\left(\frac{k}{n}\right)=(2\pi)^{% (n-1)/2}n^{-1/2}.$
 5.5.8 $\mathop{\psi\/}\nolimits\!\left(2z\right)=\tfrac{1}{2}\left(\mathop{\psi\/}% \nolimits\!\left(z\right)+\mathop{\psi\/}\nolimits\!\left(z+\tfrac{1}{2}\right% )\right)+\mathop{\ln\/}\nolimits 2,$ Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 5.5.9 $\mathop{\psi\/}\nolimits\!\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}\mathop{% \psi\/}\nolimits\!\left(z+\frac{k}{n}\right)+\mathop{\ln\/}\nolimits n.$

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