5 Gamma Function Properties 5.3 Graphics 5.5 Functional Relations

§5.4 Special Values and Extrema

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Keywords:: gamma function, special values
Permalink:: http://dlmf.nist.gov/5.4
See also:: Annotations for Ch.5

§5.4(i) Gamma Function

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Notes:: For (5.4.1), (5.4.3)–(5.4.6) and (5.4.11) see Olver (1997b, pp.32, 35, 38, and 39). For (5.4.2) use (5.4.6) and (5.5.1).
Permalink:: http://dlmf.nist.gov/5.4.i
See also:: Annotations for §5.4 and Ch.5

5.4.1	$\displaystyle\Gamma\left(1\right)$	$\displaystyle=1,$
5.4.1	$\displaystyle n!$	$\displaystyle=\Gamma\left(n+1\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer Referenced by: §5.4(i), How to Cite Permalink: http://dlmf.nist.gov/5.4.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.4(i), §5.4 and Ch.5

5.4.2		$n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!!$ : double factorial and $n$ : nonnegative integer Referenced by: §5.4(i), §5.4(i) Permalink: http://dlmf.nist.gov/5.4.E2 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5

(The second line of Formula (5.4.2) also applies when $n=-1$ .)

5.4.3		$\|\Gamma\left(iy\right)\|=\left(\frac{\pi}{y\sinh\left(\pi y\right)}\right)^{1/2},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.1.29 Referenced by: §10.24, §5.4(i) Permalink: http://dlmf.nist.gov/5.4.E3 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5

5.4.4		$\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\Gamma\left(\tfrac{1}{2}-\mathrm{i}% y\right)=\left\|\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\right\|^{2}=\frac{% \pi}{\cosh\left(\pi y\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.1.30 Permalink: http://dlmf.nist.gov/5.4.E4 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5

5.4.5		$\Gamma\left(\tfrac{1}{4}+\mathrm{i}y\right)\Gamma\left(\tfrac{3}{4}-\mathrm{i}% y\right)=\frac{\pi\sqrt{2}}{\cosh\left(\pi y\right)+\mathrm{i}\sinh\left(\pi y% \right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.1.32 Permalink: http://dlmf.nist.gov/5.4.E5 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5

5.4.6	$\displaystyle\Gamma\left(\tfrac{1}{2}\right)$	$\displaystyle=\pi^{1/2}=1.77245\;38509\;05516\;02729\;\dots,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function and $\pi$ : the ratio of the circumference of a circle to its diameter A&S Ref: 6.1.8 Notes: For more digits see OEIS Sequence A002161; see also Sloane (2003). Referenced by: §4.10(ii), §5.4(i) Permalink: http://dlmf.nist.gov/5.4.E6 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5
5.4.7	$\displaystyle\Gamma\left(\tfrac{1}{3}\right)$	$\displaystyle=2.67893\;85347\;07747\;63365\;\dots,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function A&S Ref: 6.1.11 (where the value is computed to 10D.) Notes: For more digits see OEIS Sequence A073005; see also Sloane (2003). Permalink: http://dlmf.nist.gov/5.4.E7 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5
5.4.8	$\displaystyle\Gamma\left(\tfrac{2}{3}\right)$	$\displaystyle=1.35411\;79394\;26400\;41694\;\dots,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function A&S Ref: 6.1.13 (where the value is computed to 10D.) Notes: For more digits see OEIS Sequence A073006; see also Sloane (2003). Permalink: http://dlmf.nist.gov/5.4.E8 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5
5.4.9	$\displaystyle\Gamma\left(\tfrac{1}{4}\right)$	$\displaystyle=3.62560\;99082\;21908\;31193\;\dots,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function A&S Ref: 6.1.10 (where the value is computed to 10D.) Notes: For more digits see OEIS Sequence A068466; see also Sloane (2003). Permalink: http://dlmf.nist.gov/5.4.E9 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5
5.4.10	$\displaystyle\Gamma\left(\tfrac{3}{4}\right)$	$\displaystyle=1.22541\;67024\;65177\;64512\;\dots.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function A&S Ref: 6.1.14 (where the value is computed to 10D.) Notes: For more digits see OEIS Sequence A068465; see also Sloane (2003). Permalink: http://dlmf.nist.gov/5.4.E10 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5
5.4.11	$\displaystyle\Gamma'\left(1\right)$	$\displaystyle=-\gamma.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function and $\gamma$ : Euler’s constant A&S Ref: 6.3.2 Referenced by: §5.4(i) Permalink: http://dlmf.nist.gov/5.4.E11 Encodings: TeX, pMML, png See also: Annotations for §5.4(i), §5.4 and Ch.5

§5.4(ii) Psi Function

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Keywords:: psi function, special values
Notes:: For (5.4.12)–(5.4.18) see Olver (1997b, p. 39) and for (5.4.19) see Andrews et al. (1999, p. 15).
Permalink:: http://dlmf.nist.gov/5.4.ii
See also:: Annotations for §5.4 and Ch.5

5.4.12	$\displaystyle\psi\left(1\right)$	$\displaystyle=-\gamma,$
5.4.12	$\displaystyle\psi'\left(1\right)$	$\displaystyle=\tfrac{1}{6}\pi^{2},$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter and $\psi\left(\NVar{z}\right)$ : psi (or digamma) function A&S Ref: 6.3.2 Referenced by: §5.4(ii) Permalink: http://dlmf.nist.gov/5.4.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

5.4.13	$\displaystyle\psi\left(\tfrac{1}{2}\right)$	$\displaystyle=-\gamma-2\ln 2,$
5.4.13	$\displaystyle\psi'\left(\tfrac{1}{2}\right)$	$\displaystyle=\tfrac{1}{2}\pi^{2}.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 6.3.3 Referenced by: §5.19(i) Permalink: http://dlmf.nist.gov/5.4.E13 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

For higher derivatives of $\psi\left(z\right)$ at $z=1$ and $z=\frac{1}{2}$ , see §5.15.

5.4.14		$\psi\left(n+1\right)=\sum_{k=1}^{n}\frac{1}{k}-\gamma,$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $n$ : nonnegative integer and $k$ : nonnegative integer A&S Ref: 6.3.2 Permalink: http://dlmf.nist.gov/5.4.E14 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

5.4.15		$\psi\left(n+\tfrac{1}{2}\right)=-\gamma-2\ln 2+2\left(1+\tfrac{1}{3}+\dots+% \tfrac{1}{2n-1}\right),$
$n=1,2,\dots$ .
ⓘ Symbols: $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function and $n$ : nonnegative integer A&S Ref: 6.3.4 Permalink: http://dlmf.nist.gov/5.4.E15 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

5.4.16		$\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.3.11 Permalink: http://dlmf.nist.gov/5.4.E16 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

5.4.17		$\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.3.12 Permalink: http://dlmf.nist.gov/5.4.E17 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

5.4.18		$\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit and $y$ : real variable A&S Ref: 6.3.13 Referenced by: §5.4(ii) Permalink: http://dlmf.nist.gov/5.4.E18 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

If $p, q$ are integers with $0<p<q$ , then

5.4.19		$\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}% {q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln% \left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function, $q$ : real or complex variable and $k$ : nonnegative integer Referenced by: §5.19(i), §5.4(ii) Permalink: http://dlmf.nist.gov/5.4.E19 Encodings: TeX, pMML, png See also: Annotations for §5.4(ii), §5.4 and Ch.5

§5.4(iii) Extrema

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Keywords:: asymptotic approximation, extrema, gamma function, psi function, table of, zeros
Notes:: For (5.4.20) use (5.11.2) to solve $\psi\left(1-x\right)=\pi\cot\left(\pi x\right)$ with $x=-n+u$ and $n$ large.
Referenced by:: Figure 5.3.1, Figure 5.3.1
Permalink:: http://dlmf.nist.gov/5.4.iii
See also:: Annotations for §5.4 and Ch.5

Table 5.4.1:

\Gamma'\left(x_{n}\right)=\psi\left(x_{n}\right)=0

$n$	$x_{n}$	$\Gamma\left(x_{n}\right)$
0	$1.46163\;21449\;68362\;34126$	$0.88560\;31944\;10888\;70028$
1	$-0.50408\;30082\;64455\;40926$	$-3.54464\;36111\;55005\;08912$
2	$-1.57349\;84731\;62390\;45878$	$2.30240\;72583\;39680\;13582$
3	$-2.61072\;08684\;44144\;65000$	$-0.88813\;63584\;01241\;92010$
4	$-3.63529\;33664\;36901\;09784$	$0.24512\;75398\;34366\;25044$
5	$-4.65323\;77617\;43142\;44171$	$-0.05277\;96395\;87319\;40076$
6	$-5.66716\;24415\;56885\;53585$	$0.00932\;45944\;82614\;85052$
7	$-6.67841\;82130\;73426\;74283$	$-0.00139\;73966\;08949\;76730$
8	$-7.68778\;83250\;31626\;03744$	$0.00018\;18784\;44909\;40419$
9	$-8.69576\;41638\;16401\;26649$	$-0.00002\;09252\;90446\;52667$
10	$-9.70267\;25400\;01863\;73608$	$0.00000\;21574\;16104\;52285$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: §6.3 ((Values are given to 3 decimals for $n=0,1,\dots,7$ .))
Referenced by:: How to Cite, Erratum (V1.0.18) for Table 5.4.1
Permalink:: http://dlmf.nist.gov/5.4.T1
Correction/Addition (effective with 1.0.18):: The table of extrema for $\Gamma$ had several entries in the $x_{n}$ column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.
Suggested 2018-02-17 by David Smith
See also:: Annotations for §5.4(iii), §5.4 and Ch.5

Compare Figure 5.3.1.

As $n\to\infty$ ,

5.4.20		$x_{n}=-n+\frac{1}{\pi}\operatorname{arctan}\left(\frac{\pi}{\ln n}\right)+O% \left(\frac{1}{n(\ln n)^{2}}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 6.3.20 (The error estimate has been improved.) Referenced by: §5.4(iii) Permalink: http://dlmf.nist.gov/5.4.E20 Encodings: TeX, pMML, png See also: Annotations for §5.4(iii), §5.4 and Ch.5

For error bounds for this estimate see Walker (2007, Theorem 5).

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§5.4 Special Values and Extrema

Contents

§5.4(i) Gamma Function

§5.4(ii) Psi Function

§5.4(iii) Extrema