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
§5.4(ii) Psi Function
§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

$\Gamma\left(1\right)=1,$

$n!=\Gamma\left(n+1\right).$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : factorial (as in ) and : 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 $n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, : double factorial and : 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 .)

5.4.3 $|\Gamma\left(iy\right)|=\left(\frac{\pi}{y\sinh\left(\pi y\right)}\right)^{1/2},$

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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 : 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 $\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\Gamma\left(\tfrac{1}{2}-\mathrm{i}% y\right)=\left|\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\right|^{2}=\frac{% \pi}{\cosh\left(\pi y\right)},$

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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 : 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 $\Gamma\left(\tfrac{1}{4}+\mathrm{i}y\right)\Gamma\left(\tfrac{3}{4}-\mathrm{i}% y\right)=\frac{\pi\sqrt{2}}{\cosh\left(\pi y\right)+\mathrm{i}\sinh\left(\pi y% \right)}.$

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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 : 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 $\Gamma\left(\tfrac{1}{2}\right)=\pi^{1/2}\\ =1.77245\;38509\;05516\;02729\;\dots,$

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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 $\Gamma\left(\tfrac{1}{3}\right)=2.67893\;85347\;07747\;63365\;\dots,$

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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 $\Gamma\left(\tfrac{2}{3}\right)=1.35411\;79394\;26400\;41694\;\dots,$

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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 $\Gamma\left(\tfrac{1}{4}\right)=3.62560\;99082\;21908\;31193\;\dots,$

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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 $\Gamma\left(\tfrac{3}{4}\right)=1.22541\;67024\;65177\;64512\;\dots.$

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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 $\Gamma'\left(1\right)=-\gamma.$

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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

$\psi\left(1\right)=-\gamma,$

$\psi'\left(1\right)=\tfrac{1}{6}\pi^{2},$

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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

$\psi\left(\tfrac{1}{2}\right)=-\gamma-2\ln 2,$

$\psi'\left(\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}.$

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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 and $z=\frac{1}{2}$ , see §5.15.

5.4.14 $\psi\left(n+1\right)=\sum_{k=1}^{n}\frac{1}{k}-\gamma,$

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Symbols:: $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : nonnegative integer and : 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 $\psi\left(n+\tfrac{1}{2}\right)=-\gamma-2\ln 2+2\left(1+\tfrac{1}{3}+\dots+% \tfrac{1}{2n-1}\right),$ $n=1,2,\dots$ .

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Symbols:: $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$

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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 : 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 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$

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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 : 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 $\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right).$

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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 : 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 are integers with , then

5.4.19 $\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}% {q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln% \left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).$

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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, : real or complex variable and : 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 and 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: $\Gamma'\left(x_{n}\right)=\psi\left(x_{n}\right)=0$ .

	$x_{n}$	$\Gamma\left(x_{n}\right)$
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, : nonnegative integer and : 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