§5.4 Special Values and Extrema

Keywords:: gamma function
Permalink:: http://dlmf.nist.gov/5.4

§5.4(i) Gamma Function
§5.4(ii) Psi Function
§5.4(iii) Extrema

§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

5.4.1

$\mathop{\Gamma\/}\nolimits\!\left(1\right)=1,$

$n!=\mathop{\Gamma\/}\nolimits\!\left(n+1\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial and $n$ : nonnegative integer
Referenced by:: §5.4(i)
Permalink:: http://dlmf.nist.gov/5.4.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

5.4.2 $n!!=\begin{cases}2^{{\frac{1}{2}n}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}n+1\right),&n\text{ even},\\ \pi^{{-\frac{1}{2}}}2^{{\frac{1}{2}n+\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}n+1\right),&n\text{ odd}.\end{cases}$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!!$ : $n!!$ : 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

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function 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

5.4.4 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+iy\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-iy\right)=\left|\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+iy\right)\right|^{2}=\frac{\pi}{\mathop{\cosh\/}\nolimits\!\left(\pi y\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $y$ : real variable
A&S Ref:: 6.1.30
Permalink:: http://dlmf.nist.gov/5.4.E4
Encodings:: TeX, pMML, png

5.4.5 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{4}+iy\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{4}-iy\right)=\frac{\pi\sqrt{2}}{\mathop{\cosh\/}\nolimits\!\left(\pi y\right)+i\mathop{\sinh\/}\nolimits\!\left(\pi y\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and $y$ : real variable
A&S Ref:: 6.1.32
Permalink:: http://dlmf.nist.gov/5.4.E5
Encodings:: TeX, pMML, png

5.4.6 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\right)=\pi^{{1/2}}\\ =1.77245\; 38509\; 0 5516\; 0 2729\;\dots,$

Notes:: For more digits see OEIS Sequence A002161; see also Sloane (2003).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 6.1.8
Referenced by:: §4.10(ii), §5.4(i)
Permalink:: http://dlmf.nist.gov/5.4.E6
Encodings:: TeX, pMML, png

5.4.7 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)=2.67893\; 85347\; 0 7747\; 63365\;\dots,$

Notes:: For more digits see OEIS Sequence A073005; see also Sloane (2003).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 6.1.11 (where the value is computed to 10D.)
Permalink:: http://dlmf.nist.gov/5.4.E7
Encodings:: TeX, pMML, png

5.4.8 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)=1.35411\; 79394\; 26400\; 41694\;\dots,$

Notes:: For more digits see OEIS Sequence A073006; see also Sloane (2003).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 6.1.13 (where the value is computed to 10D.)
Permalink:: http://dlmf.nist.gov/5.4.E8
Encodings:: TeX, pMML, png

5.4.9 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{4}\right)=3.62560\; 99082\; 21908\; 31193\;\dots,$

Notes:: For more digits see OEIS Sequence A068466; see also Sloane (2003).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 6.1.10 (where the value is computed to 10D.)
Permalink:: http://dlmf.nist.gov/5.4.E9
Encodings:: TeX, pMML, png

5.4.10 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{4}\right)=1.22541\; 67024\; 65177\; 64512\;\dots.$

Notes:: For more digits see OEIS Sequence A068465; see also Sloane (2003).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 6.1.14 (where the value is computed to 10D.)
Permalink:: http://dlmf.nist.gov/5.4.E10
Encodings:: TeX, pMML, png

5.4.11 ${\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(1\right)=-\EulerConstant.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\EulerConstant$ : 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

§5.4(ii) Psi Function

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).
Keywords:: psi function
Permalink:: http://dlmf.nist.gov/5.4.ii

5.4.12

$\mathop{\psi\/}\nolimits\!\left(1\right)=-\EulerConstant,$

${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(1\right)=\tfrac{1}{6}\pi^{2},$

Symbols:: $\EulerConstant$ : Euler’s constant and $\mathop{\psi\/}\nolimits\!\left(z\right)$ : 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

5.4.13

$\mathop{\psi\/}\nolimits\!\left(\tfrac{1}{2}\right)=-\EulerConstant-2\mathop{\ln\/}\nolimits 2,$

${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}.$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function and $\mathop{\ln\/}\nolimits z$ : 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

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

5.4.14 $\mathop{\psi\/}\nolimits\!\left(n+1\right)=\sum _{{k=1}}^{n}\frac{1}{k}-\EulerConstant,$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : 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

5.4.15 $\mathop{\psi\/}\nolimits\!\left(n+\tfrac{1}{2}\right)=-\EulerConstant-2\mathop{\ln\/}\nolimits 2+2\left(1+\tfrac{1}{3}+\dots+\tfrac{1}{2n-1}\right),$ $n=1,2,\dots$ .

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : 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

5.4.16 $\imagpart{\mathop{\psi\/}\nolimits\!\left(iy\right)}=\frac{1}{2y}+\frac{\pi}{2}\mathop{\coth\/}\nolimits\!\left(\pi y\right),$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\imagpart{}$ : imaginary part and $y$ : real variable
A&S Ref:: 6.3.11
Permalink:: http://dlmf.nist.gov/5.4.E16
Encodings:: TeX, pMML, png

5.4.17 $\imagpart{\mathop{\psi\/}\nolimits\!\left(\tfrac{1}{2}+iy\right)}=\frac{\pi}{2}\mathop{\tanh\/}\nolimits\!\left(\pi y\right),$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\imagpart{}$ : imaginary part and $y$ : real variable
A&S Ref:: 6.3.12
Permalink:: http://dlmf.nist.gov/5.4.E17
Encodings:: TeX, pMML, png

5.4.18 $\imagpart{\mathop{\psi\/}\nolimits\!\left(1+iy\right)}=-\frac{1}{2y}+\frac{\pi}{2}\mathop{\coth\/}\nolimits\!\left(\pi y\right).$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\imagpart{}$ : imaginary part 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

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

5.4.19 $\mathop{\psi\/}\nolimits\!\left(\frac{p}{q}\right)=-\EulerConstant-\mathop{\ln\/}\nolimits q-\frac{\pi}{2}\mathop{\cot\/}\nolimits\!\left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum _{{k=1}}^{{q-1}}\mathop{\cos\/}\nolimits\!\left(\frac{2\pi kp}{q}\right)\mathop{\ln\/}\nolimits\!\left(2-2\mathop{\cos\/}\nolimits\!\left(\frac{2\pi k}{q}\right)\right).$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : 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

§5.4(iii) Extrema

Notes:: For (5.4.20) use (5.11.2) to solve $\mathop{\psi\/}\nolimits\!\left(1-x\right)=\pi\mathop{\cot\/}\nolimits\!\left(\pi x\right)$ with $x=-n+u$ and $n$ large.
Keywords:: gamma function, psi function
Referenced by:: Figure 5.3.1, Figure 5.3.1
Permalink:: http://dlmf.nist.gov/5.4.iii

Table 5.4.1: ${\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x_{n}\right)=\mathop{\psi\/}\nolimits\!\left(x_{n}\right)=0$ .

$n$	$x_{n}$	$\mathop{\Gamma\/}\nolimits\!\left(x_{n}\right)$
0	1.46163 21449	0.88560 31944
1	−0.50408 30083	−3.54464 36112
2	−1.57349 84732	2.30240 72583
3	−2.61072 08875	−0.88813 63584
4	−3.63529 33665	0.24512 75398
5	−4.65323 77626	−0.05277 96396
6	−5.66716 24513	0.00932 45945
7	−6.67841 82649	−0.00139 73966
8	−7.68778 83250	0.00018 18784
9	−8.69576 41633	−0.00002 09253
10	−9.70267 25406	0.00000 21574

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: §6.3 ((Values are given to 3 decimals for $n=0,1,\dots,7$ .))
Permalink:: http://dlmf.nist.gov/5.4.T1

Compare Figure 5.3.1.

As $n\to\infty$ ,

5.4.20 $x_{n}=-n+\frac{1}{\pi}\mathop{\mathrm{arctan}\/}\nolimits\!\left(\frac{\pi}{\mathop{\ln\/}\nolimits n}\right)+\mathop{O\/}\nolimits\!\left(\frac{1}{n(\mathop{\ln\/}\nolimits n)^{2}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : 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

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

§5.4 Special Values and Extrema

Contents

§5.4(i) Gamma Function

§5.4(ii) Psi Function

§5.4(iii) Extrema