5 Gamma FunctionProperties5.3 Graphics5.5 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

Contents

1. §5.4(i) Gamma Function
2. §5.4(ii) Psi Function
3. §5.4(iii) Extrema

§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		$\mathrm{\Gamma }\left(1\right)$	$=1,$
		$n!$	$=\mathrm{\Gamma }\left(n+1\right).$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$|\mathrm{\Gamma }\left(\mathrm{i}y\right)|={\left(\frac{\pi }{y\sinh \left(\pi y\right)}\right)}^{1/2},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: 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		$$\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{i}y\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mathrm{i}y\right)={\left|\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{i}y\right)\right|}^2=\frac{\pi }{\cosh \left(\pi y\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: 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		$$\mathrm{\Gamma }\left(\frac{1}{4}+\mathrm{i}y\right)\mathrm{\Gamma }\left(\frac{3}{4}-\mathrm{i}y\right)=\frac{\pi \sqrt{2}}{\cosh \left(\pi y\right)+\mathrm{i}\sinh \left(\pi y\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		$\mathrm{\Gamma }\left(\frac{1}{2}\right)$	$={\pi }^{1/2}=\mathrm{1.77245\hspace{0.33em}38509\hspace{0.33em}05516\hspace{0.33em}02729}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$\mathrm{\Gamma }\left(\frac{1}{3}\right)$	$=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$\mathrm{\Gamma }\left(\frac{2}{3}\right)$	$=\mathrm{1.35411\hspace{0.33em}79394\hspace{0.33em}26400\hspace{0.33em}41694}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$\mathrm{\Gamma }\left(\frac{1}{4}\right)$	$=\mathrm{3.62560\hspace{0.33em}99082\hspace{0.33em}21908\hspace{0.33em}31193}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$\mathrm{\Gamma }\left(\frac{3}{4}\right)$	$=\mathrm{1.22541\hspace{0.33em}67024\hspace{0.33em}65177\hspace{0.33em}64512}\mathrm{\dots }.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		${\mathrm{\Gamma }}^{\prime }\left(1\right)$	$=-\gamma .$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$\psi \left(1\right)$	$=-\gamma ,$
		${\psi }^{\prime }\left(1\right)$	$=\frac{1}{6}{\pi }^2,$
ⓘ Symbols: $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter and $\psi \left(z\right)$: 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		$\psi \left(\frac{1}{2}\right)$	$=-\gamma -2\ln 2,$
		${\psi }^{\prime }\left(\frac{1}{2}\right)$	$=\frac{1}{2}{\pi }^2.$
ⓘ Symbols: $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: 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, $\psi \left(z\right)$: 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		$$\psi \left(n+\frac{1}{2}\right)=-\gamma -2\ln 2+2\left(1+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{2n-1}\right),$$
$n=1,2,\mathrm{\dots }$.
ⓘ Symbols: $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: 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		$$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: 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		$$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: 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		$$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: 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 , 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, $\cos z$: cosine function, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: 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

$n$	$x_n$	$\mathrm{\Gamma }\left(x_n\right)$
0	$\mathrm{1.46163\hspace{0.33em}21449\hspace{0.33em}68362\hspace{0.33em}34126}$	$\mathrm{0.88560\hspace{0.33em}31944\hspace{0.33em}10888\hspace{0.33em}70028}$
1	$-\mathrm{0.50408\hspace{0.33em}30082\hspace{0.33em}64455\hspace{0.33em}40926}$	$-\mathrm{3.54464\hspace{0.33em}36111\hspace{0.33em}55005\hspace{0.33em}08912}$
2	$-\mathrm{1.57349\hspace{0.33em}84731\hspace{0.33em}62390\hspace{0.33em}45878}$	$\mathrm{2.30240\hspace{0.33em}72583\hspace{0.33em}39680\hspace{0.33em}13582}$
3	$-\mathrm{2.61072\hspace{0.33em}08684\hspace{0.33em}44144\hspace{0.33em}65000}$	$-\mathrm{0.88813\hspace{0.33em}63584\hspace{0.33em}01241\hspace{0.33em}92010}$
4	$-\mathrm{3.63529\hspace{0.33em}33664\hspace{0.33em}36901\hspace{0.33em}09784}$	$\mathrm{0.24512\hspace{0.33em}75398\hspace{0.33em}34366\hspace{0.33em}25044}$
5	$-\mathrm{4.65323\hspace{0.33em}77617\hspace{0.33em}43142\hspace{0.33em}44171}$	$-\mathrm{0.05277\hspace{0.33em}96395\hspace{0.33em}87319\hspace{0.33em}40076}$
6	$-\mathrm{5.66716\hspace{0.33em}24415\hspace{0.33em}56885\hspace{0.33em}53585}$	$\mathrm{0.00932\hspace{0.33em}45944\hspace{0.33em}82614\hspace{0.33em}85052}$
7	$-\mathrm{6.67841\hspace{0.33em}82130\hspace{0.33em}73426\hspace{0.33em}74283}$	$-\mathrm{0.00139\hspace{0.33em}73966\hspace{0.33em}08949\hspace{0.33em}76730}$
8	$-\mathrm{7.68778\hspace{0.33em}83250\hspace{0.33em}31626\hspace{0.33em}03744}$	$\mathrm{0.00018\hspace{0.33em}18784\hspace{0.33em}44909\hspace{0.33em}40419}$
9	$-\mathrm{8.69576\hspace{0.33em}41638\hspace{0.33em}16401\hspace{0.33em}26649}$	$-\mathrm{0.00002\hspace{0.33em}09252\hspace{0.33em}90446\hspace{0.33em}52667}$
10	$-\mathrm{9.70267\hspace{0.33em}25400\hspace{0.33em}01863\hspace{0.33em}73608}$	$\mathrm{0.00000\hspace{0.33em}21574\hspace{0.33em}16104\hspace{0.33em}52285}$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

§6.3 ((Values are given to 3 decimals for $n=0,1,\mathrm{\dots },7$.))

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 $\mathrm{\Gamma }$ 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 $n\to \mathrm{\infty }$,

5.4.20		$$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: 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 Graphics5.5 Functional Relations

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