# §5.4 Special Values and Extrema

## §5.4(i) Gamma Function

 5.4.1 $\displaystyle\Gamma\left(1\right)$ $\displaystyle=1,$ $\displaystyle n!$ $\displaystyle=\Gamma\left(n+1\right).$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$) and $n$: nonnegative integer Referenced by: §5.4(i) 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 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 (as in $n!!$) 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 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 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 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 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 5
 5.4.5 $\Gamma\left(\tfrac{1}{4}+\mathrm{i}y\right)\Gamma\left(\tfrac{3}{4}-\mathrm{i}% y\right)=\frac{\pi\sqrt{2}}{\cosh\left(\pi y\right)+\mathrm{i}\sinh\left(\pi y% \right)}.$
 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 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 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 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 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 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 5

## §5.4(ii) Psi Function

 5.4.12 $\displaystyle\psi\left(1\right)$ $\displaystyle=-\gamma,$ $\displaystyle\psi'\left(1\right)$ $\displaystyle=\tfrac{1}{6}\pi^{2},$ ⓘ 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 5
 5.4.13 $\displaystyle\psi\left(\tfrac{1}{2}\right)$ $\displaystyle=-\gamma-2\ln 2,$ $\displaystyle\psi'\left(\tfrac{1}{2}\right)$ $\displaystyle=\tfrac{1}{2}\pi^{2}.$ ⓘ 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 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 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 5
 5.4.16 $\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$
 5.4.17 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$
 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 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 5

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

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

## §5.4(iii) Extrema

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 5

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