§ 5.5. Functional Relations

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§5.5(i)Recurrence
§5.5(ii)Reflection
§5.5(iii)Multiplication
§5.5(iv)Bohr-Mollerup Theorem

§ 5.5(i). Recurrence

Notes:: See Olver (1997b, pp. 32 and 39), or Temme (1996, pp. 42 and 54).
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5.5.1 $\Gamma\!\left(z+1\right)=z\Gamma\!\left(z\right),$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.15
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5.5.2 $\psi\!\left(z+1\right)=\psi\!\left(z\right)+\frac{1}{z}.$

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.5
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§ 5.5(ii). Reflection

Notes:: See Olver (1997b, pp. 35 and 39).
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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:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.17 (without the condition on $z$ .)
Referenced by:: §5.21, §9.12(vi)
Permalink:: http://dlmf.nist.gov/5.5.E3
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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:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.7 (without the condition on $z$ .)
Permalink:: http://dlmf.nist.gov/5.5.E4
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§ 5.5(iii). Multiplication

Notes:: For (5.5.5) see Olver (1997b, p. 35); for (5.5.6)–(5.5.8) see Temme (1996, pp. 52–53 and 76). (5.5.9) follows from (5.5.6).
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¶ 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:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.18
Referenced by:: §5.5(iii)
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¶ 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:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.20
Referenced by:: §5.5(iii)
Permalink:: http://dlmf.nist.gov/5.5.E6
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5.5.7 $\prod _{{k=1}}^{{n-1}}\Gamma\!\left(\frac{k}{n}\right)=(2\pi)^{{(n-1)/2}}n^{{-1/2}}.$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer and $k$ : nonnegative integer
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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:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.8
Referenced by:: §5.5(iii)
Permalink:: http://dlmf.nist.gov/5.5.E8
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5.5.9 $\psi\!\left(nz\right)=\frac{1}{n}\sum _{{k=0}}^{{n-1}}\psi\!\left(z+\frac{k}{n}\right)+\ln n.$

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.5(iii)
Permalink:: http://dlmf.nist.gov/5.5.E9
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§ 5.5(iv). Bohr-Mollerup Theorem

Notes:: See Andrews et al. (1999, pp. 34–36).
Referenced by:: §5.18(ii), Fig.5.3.2, Fig.5.3.2
Permalink:: http://dlmf.nist.gov/5.5.SS4

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 §Ch.1), then $f(x)=\Gamma\!\left(x\right)$ .