5 Gamma Function Properties 5.3 Graphics 5.5 Functional Relations

§5.4 Special Values and Extrema

ⓘ

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

§5.4(i) Gamma Function

ⓘ

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

ⓘ

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

ⓘ

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	$-9.70267\;25400\;01863\;73608$	$0.00000\;21574\;16104\;52285$

ⓘ

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).

5.3 Graphics 5.5 Functional Relations

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

§5.4 Special Values and Extrema

Contents

§5.4(i) Gamma Function

§5.4(ii) Psi Function

§5.4(iii) Extrema