Particular values of the gamma function

Template:Short description The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

Integers and half-integers

For positive integer arguments, the gamma function coincides with the factorial. That is,

$\mathrm{\Gamma }(n)=(n-1)!,$

and hence

$\begin{array}{cc}\mathrm{\Gamma }(1)& =1,\\ \mathrm{\Gamma }(2)& =1,\\ \mathrm{\Gamma }(3)& =2,\\ \mathrm{\Gamma }(4)& =6,\\ \mathrm{\Gamma }(5)& =24,\end{array}$

and so on. For non-positive integers, the gamma function is not defined.

For positive half-integers, the function values are given exactly by

$\mathrm{\Gamma }\left(\frac{n}{2}\right)=\sqrt{\pi }\frac{(n-2)!!}{2^{\frac{n-1}{2}}},$

or equivalently, for non-negative integer values of Template:Mvar:

$\begin{array}{cc}\mathrm{\Gamma }(\frac{1}{2}+n)& =\frac{(2n-1)!!}{2^n}\sqrt{\pi }=\frac{(2n)!}{4^{n}n!}\sqrt{\pi }\\ \mathrm{\Gamma }(\frac{1}{2}-n)& =\frac{(-2{)}^n}{(2n-1)!!}\sqrt{\pi }=\frac{(-4{)}^{n}n!}{(2n)!}\sqrt{\pi }\end{array}$

where Template:Math denotes the double factorial. In particular,

$\mathrm{\Gamma }\left(\frac{1}{2}\right)$	$=\sqrt{\pi }$	$\approx 1.7724538509055160273,$	Template:OEIS2C
$\mathrm{\Gamma }\left(\frac{3}{2}\right)$	$=\frac{1}{2}\sqrt{\pi }$	$\approx 0.8862269254527580137,$	Template:OEIS2C
$\mathrm{\Gamma }\left(\frac{5}{2}\right)$	$=\frac{3}{4}\sqrt{\pi }$	$\approx 1.3293403881791370205,$	Template:OEIS2C
$\mathrm{\Gamma }\left(\frac{7}{2}\right)$	$=\frac{15}{8}\sqrt{\pi }$	$\approx 3.3233509704478425512,$	Template:OEIS2C

and by means of the reflection formula,

$\mathrm{\Gamma }(-\frac{1}{2})$	$=-2\sqrt{\pi }$	$\approx -3.5449077018110320546,$	Template:OEIS2C
$\mathrm{\Gamma }(-\frac{3}{2})$	$=\frac{4}{3}\sqrt{\pi }$	$\approx 2.3632718012073547031,$	Template:OEIS2C
$\mathrm{\Gamma }(-\frac{5}{2})$	$=-\frac{8}{15}\sqrt{\pi }$	$\approx -0.9453087204829418812,$	Template:OEIS2C

General rational argument

In analogy with the half-integer formula,

$\begin{array}{cc}\mathrm{\Gamma }(n+\frac{1}{3})& =\mathrm{\Gamma }\left(\frac{1}{3}\right)\frac{(3n-2)!!!}{3^n}\\ \mathrm{\Gamma }(n+\frac{1}{4})& =\mathrm{\Gamma }\left(\frac{1}{4}\right)\frac{(4n-3)!!!!}{4^n}\\ \mathrm{\Gamma }(n+\frac{1}{q})& =\mathrm{\Gamma }\left(\frac{1}{q}\right)\frac{(qn-(q-1)){!}^{(q)}}{q^n}\\ \mathrm{\Gamma }(n+\frac{p}{q})& =\mathrm{\Gamma }\left(\frac{p}{q}\right)\frac{1}{q^n}\prod _{k=1}^n(kq+p-q)\end{array}$

where Template:Math denotes the Template:Mvarth multifactorial of Template:Mvar. Numerically,

$\mathrm{\Gamma }\left(\frac{1}{3}\right)\approx 2.6789385347077476337$ Template:OEIS2C

$\mathrm{\Gamma }\left(\frac{1}{4}\right)\approx 3.6256099082219083119$ Template:OEIS2C

$\mathrm{\Gamma }\left(\frac{1}{5}\right)\approx 4.5908437119988030532$ Template:OEIS2C

$\mathrm{\Gamma }\left(\frac{1}{6}\right)\approx 5.5663160017802352043$ Template:OEIS2C

$\mathrm{\Gamma }\left(\frac{1}{7}\right)\approx 6.5480629402478244377$ Template:OEIS2C

$\mathrm{\Gamma }\left(\frac{1}{8}\right)\approx 7.5339415987976119047$ Template:OEIS2C.

As $n$ tends to infinity,

$\mathrm{\Gamma }\left(\frac{1}{n}\right)\sim n-\gamma$

where $\gamma$ is the Euler–Mascheroni constant and $\sim$ denotes asymptotic equivalence.

It is unknown whether these constants are transcendental in general, but Template:Math and Template:Math were shown to be transcendental by G. V. Chudnovsky. Template:Math has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Template:Math, Template:Math, and Template:Math are algebraically independent.

For $n\ge 2$ at least one of the two numbers $\mathrm{\Gamma }\left(\frac{1}{n}\right)$ and $\mathrm{\Gamma }\left(\frac{2}{n}\right)$ is transcendental.

The number $\mathrm{\Gamma }\left(\frac{1}{4}\right)$ is related to the lemniscate constant Template:Mvar by

$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\sqrt{2\varpi \sqrt{2\pi }}$

Borwein and Zucker have found that Template:Math can be expressed algebraically in terms of Template:Mvar, Template:Math, Template:Math, Template:Math, and Template:Math where Template:Math is a complete elliptic integral of the first kind. This permits efficiently approximating the gamma function of rational arguments to high precision using quadratically convergent arithmetic–geometric mean iterations. For example:

$\begin{array}{cc}\mathrm{\Gamma }\left(\frac{1}{6}\right)& =\frac{\sqrt{\frac{3}{\pi }}\mathrm{\Gamma }{\left(\frac{1}{3}\right)}^2}{\sqrt[3]{2}}\\ \mathrm{\Gamma }\left(\frac{1}{4}\right)& =2\sqrt{K\left(\frac{1}{2}\right)\sqrt{\pi }}\\ \mathrm{\Gamma }\left(\frac{1}{3}\right)& =\frac{2^{7/9}\sqrt[3]{\pi K\left(\frac{1}{4}(2-\sqrt{3})\right)}}{\sqrt[12]{3}}\\ \mathrm{\Gamma }\left(\frac{1}{8}\right)\mathrm{\Gamma }\left(\frac{3}{8}\right)& =8\sqrt[4]{2}\sqrt{(\sqrt{2}-1)\pi }K(3-2\sqrt{2})\\ \frac{\mathrm{\Gamma }\left(\frac{1}{8}\right)}{\mathrm{\Gamma }\left(\frac{3}{8}\right)}& =\frac{2\sqrt{(1+\sqrt{2})K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}\end{array}$

No similar relations are known for Template:Math or other denominators.

In particular, where AGM() is the arithmetic–geometric mean, we have

$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\frac{2^{\frac{7}{9}}\cdot {\pi }^{\frac{2}{3}}}{3^{\frac{1}{12}}\cdot \mathrm{AGM}{(2,\sqrt{2+\sqrt{3}})}^{\frac{1}{3}}}$

$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\sqrt{\frac{(2\pi {)}^{\frac{3}{2}}}{\mathrm{AGM}(\sqrt{2},1)}}$

$\mathrm{\Gamma }\left(\frac{1}{6}\right)=\frac{2^{\frac{14}{9}}\cdot 3^{\frac{1}{3}}\cdot {\pi }^{\frac{5}{6}}}{\mathrm{AGM}{(1+\sqrt{3},\sqrt{8})}^{\frac{2}{3}}}.$

Other formulas include the infinite products

$\mathrm{\Gamma }\left(\frac{1}{4}\right)=(2\pi {)}^{\frac{3}{4}}\prod _{k=1}^{\mathrm{\infty }}\tanh \left(\frac{\pi k}{2}\right)$

and

$\mathrm{\Gamma }\left(\frac{1}{4}\right)=A^3{e}^{-\frac{G}{\pi }}\sqrt{\pi }{2}^{\frac{1}{6}}\prod _{k=1}^{\mathrm{\infty }}{(1-\frac{1}{2k})}^{k(-1{)}^k}$

where Template:Mvar is the Glaisher–Kinkelin constant and Template:Mvar is Catalan's constant.

The following two representations for Template:Math were given by I. Mező

$\sqrt{\frac{\pi \sqrt{e^{\pi }}}{2}}\frac{1}{\mathrm{\Gamma }{\left(\frac{3}{4}\right)}^2}=i\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pi (k-2k^2)}{\theta }_1(\frac{i\pi }{2}(2k-1),e^{-\pi }),$

and

$\sqrt{\frac{\pi }{2}}\frac{1}{\mathrm{\Gamma }{\left(\frac{3}{4}\right)}^2}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\theta }_4(ik\pi ,e^{-\pi })}{e^{2\pi k^2}},$

where Template:Math and Template:Math are two of the Jacobi theta functions.

There also exist a number of Malmsten integrals for certain values of the gamma function:

$\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln t}{1+t^2}=\frac{\pi }{4}(2\ln 2+3\ln \pi -4\mathrm{\Gamma }\left(\frac{1}{4}\right))$

$\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln t}{1+t+t^2}=\frac{\pi }{6\sqrt{3}}(8\ln 2-3\ln 3+8\ln \pi -12\mathrm{\Gamma }\left(\frac{1}{3}\right))$

Products

Some product identities include:

$\prod _{r=1}^2\mathrm{\Gamma }\left(\frac{r}{3}\right)=\frac{2\pi }{\sqrt{3}}\approx 3.6275987284684357012$ Template:OEIS2C

$\prod _{r=1}^3\mathrm{\Gamma }\left(\frac{r}{4}\right)=\sqrt{2{\pi }^3}\approx 7.8748049728612098721$ Template:OEIS2C

$\prod _{r=1}^4\mathrm{\Gamma }\left(\frac{r}{5}\right)=\frac{4{\pi }^2}{\sqrt{5}}\approx 17.6552850814935242483$

$\prod _{r=1}^5\mathrm{\Gamma }\left(\frac{r}{6}\right)=4\sqrt{\frac{{\pi }^5}{3}}\approx 40.3993191220037900785$

$\prod _{r=1}^6\mathrm{\Gamma }\left(\frac{r}{7}\right)=\frac{8{\pi }^3}{\sqrt{7}}\approx 93.7541682035825037970$

$\prod _{r=1}^7\mathrm{\Gamma }\left(\frac{r}{8}\right)=4\sqrt{{\pi }^7}\approx 219.8287780169572636207$

In general:

$\prod _{r=1}^n\mathrm{\Gamma }\left(\frac{r}{n+1}\right)=\sqrt{\frac{(2\pi {)}^n}{n+1}}$

From those products can be deduced other values, for example, from the former equations for $\prod _{r=1}^3\mathrm{\Gamma }\left(\frac{r}{4}\right)$, $\mathrm{\Gamma }\left(\frac{1}{4}\right)$ and $\mathrm{\Gamma }\left(\frac{2}{4}\right)$, can be deduced:

$\mathrm{\Gamma }\left(\frac{3}{4}\right)={\left(\frac{\pi }{2}\right)}^{\frac{1}{4}}{\mathrm{AGM}(\sqrt{2},1)}^{\frac{1}{2}}$

Other rational relations include

$\frac{\mathrm{\Gamma }\left(\frac{1}{5}\right)\mathrm{\Gamma }\left(\frac{4}{15}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)\mathrm{\Gamma }\left(\frac{2}{15}\right)}=\frac{\sqrt{2}\sqrt[20]{3}}{\sqrt[6]{5}\sqrt[4]{5-\frac{7}{\sqrt{5}}+\sqrt{6-\frac{6}{\sqrt{5}}}}}$

$\frac{\mathrm{\Gamma }\left(\frac{1}{20}\right)\mathrm{\Gamma }\left(\frac{9}{20}\right)}{\mathrm{\Gamma }\left(\frac{3}{20}\right)\mathrm{\Gamma }\left(\frac{7}{20}\right)}=\frac{\sqrt[4]{5}(1+\sqrt{5})}{2}$

$\frac{\mathrm{\Gamma }{\left(\frac{1}{5}\right)}^2}{\mathrm{\Gamma }\left(\frac{1}{10}\right)\mathrm{\Gamma }\left(\frac{3}{10}\right)}=\frac{\sqrt{1+\sqrt{5}}}{2^{\frac{7}{10}}\sqrt[4]{5}}$

and many more relations for Template:Math where the denominator d divides 24 or 60.

Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator.

A more sophisticated example:

$\frac{\mathrm{\Gamma }\left(\frac{11}{42}\right)\mathrm{\Gamma }\left(\frac{2}{7}\right)}{\mathrm{\Gamma }\left(\frac{1}{21}\right)\mathrm{\Gamma }\left(\frac{1}{2}\right)}=\frac{8\sin \left(\frac{\pi }{7}\right)\sqrt{\sin \left(\frac{\pi }{21}\right)\sin \left(\frac{4\pi }{21}\right)\sin \left(\frac{5\pi }{21}\right)}}{2^{\frac{1}{42}}{3}^{\frac{9}{28}}{7}^{\frac{1}{3}}}$

Imaginary and complex arguments

The gamma function at the imaginary unit Template:Math gives Template:OEIS2C, Template:OEIS2C:

$\mathrm{\Gamma }(i)=(-1+i)!\approx -0.1549-0.4980i.$

It may also be given in terms of the [[Barnes G-function|Barnes Template:Mvar-function]]:

$\mathrm{\Gamma }(i)=\frac{G(1+i)}{G(i)}=e^{-\log G(i)+\log G(1+i)}.$

Curiously enough, $\mathrm{\Gamma }(i)$ appears in the below integral evaluation:

$\underset{0}{\overset{\pi /2}{\int }}\{\cot (x)\}dx=1-\frac{\pi }{2}+\frac{i}{2}\log \left(\frac{\pi }{\sinh (\pi )\mathrm{\Gamma }(i{)}^2}\right).$

Here $\{\cdot \}$ denotes the fractional part.

Because of the Euler Reflection Formula, and the fact that $\mathrm{\Gamma }(\overline{z})=\overline{\mathrm{\Gamma }}(z)$, we have an expression for the modulus squared of the Gamma function evaluated on the imaginary axis:

${|\mathrm{\Gamma }(i\kappa )|}^2=\frac{\pi }{\kappa \sinh (\pi \kappa )}$

The above integral therefore relates to the phase of $\mathrm{\Gamma }(i)$.

The gamma function with other complex arguments returns

$\mathrm{\Gamma }(1+i)=i\mathrm{\Gamma }(i)\approx 0.498-0.155i$

$\mathrm{\Gamma }(1-i)=-i\mathrm{\Gamma }(-i)\approx 0.498+0.155i$

$\mathrm{\Gamma }(\frac{1}{2}+\frac{1}{2}i)\approx 0.8181639995-0.7633138287i$

$\mathrm{\Gamma }(\frac{1}{2}-\frac{1}{2}i)\approx 0.8181639995+0.7633138287i$

$\mathrm{\Gamma }(5+3i)\approx 0.0160418827-9.4332932898i$

$\mathrm{\Gamma }(5-3i)\approx 0.0160418827+9.4332932898i.$

Other constants

The gamma function has a local minimum on the positive real axis

$x_{\min }=1.4616321449683623412626595423\dots$ Template:OEIS2C

with the value

$\mathrm{\Gamma }\left(x_{\min }\right)=0.8856031944108887002788159005\dots$ Template:OEIS2C.

Integrating the reciprocal gamma function along the positive real axis also gives the Fransén–Robinson constant.

On the negative real axis, the first local maxima and minima (zeros of the digamma function) are:

Approximate local extrema of Template:Math

Template:Mvar	Template:Math	OEIS
Template:Val	Template:Val	Template:OEIS2C
Template:Val	Template:0Template:Val	Template:OEIS2C
Template:Val	Template:Val	Template:OEIS2C
Template:Val	Template:0Template:Val	Template:OEIS2C
Template:Val	Template:Val	Template:OEIS2C
Template:Val	Template:0Template:Val	Template:OEIS2C
Template:Val	Template:Val	Template:OEIS2C
Template:Val	Template:0Template:Val	Template:OEIS2C
Template:Val	Template:Val	Template:OEIS2C
Template:Val	Template:0Template:Val	Template:OEIS2C

The only values of Template:Math for which Template:Math are Template:Math and Template:Math... Template:OEIS2C.

See also

• Chowla–Selberg formula

Further reading

• Template:Cite journal
• Template:Cite journal
• X. Gourdon & P. Sebah. Introduction to the Gamma Function
• Template:MathWorld
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal