Particular values of the Gamma function

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,

\Gamma (n)=(n-1)!\qquad n\in {\mathbb {N}}_{0},

and hence

\begin{align} \Gamma(1) &= 1, \\ \Gamma(2) &= 1, \\ \Gamma(3) &= 2, \\ \Gamma(4) &= 6, \\ \Gamma(5) &= 24. \end{align}

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

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

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

or equivalently, for non-negative integer values of $n$ :

{\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}\\\Gamma \left({\tfrac {1}{2}}-n\right)&={\frac {(-2)^{n}}{(2n-1)!!}}\,{\sqrt {\pi }}={\frac {(-4)^{n}n!}{(2n)!}}{\sqrt {\pi }}\end{aligned}}

where $n!!$ denotes the double factorial. In particular,

$\Gamma \left({\tfrac {1}{2}}\right)\,$	$= \sqrt{\pi}\,$	$\approx 1.772\,453\,850\,905\,516\,0273\,,$	A002161
$\Gamma \left({\tfrac {3}{2}}\right)\,$	$={\tfrac {1}{2}}{\sqrt {\pi }}\,$	$\approx 0.886\,226\,925\,452\,758\,0137\,,$	A019704
$\Gamma \left({\tfrac {5}{2}}\right)\,$	$={\tfrac {3}{4}}{\sqrt {\pi }}\,$	$\approx 1.329\,340\,388\,179\,137\,0205\,,$	A245884
$\Gamma \left({\tfrac {7}{2}}\right)\,$	$={\tfrac {15}{8}}{\sqrt {\pi }}\,$	$\approx 3.323\,350\,970\,447\,842\,5512\,,$	A245885

and by means of the reflection formula,

$\Gamma \left(-{\tfrac {1}{2}}\right)\,$	$= -2\sqrt{\pi}\,$	$\approx -3.544\,907\,701\,811\,032\,0546\,,$	A019707
$\Gamma \left(-{\tfrac {3}{2}}\right)\,$	$={\tfrac {4}{3}}{\sqrt {\pi }}\,$	$\approx 2.363\,271\,801\,207\,354\,7031\,,$	A245886
$\Gamma \left(-{\tfrac {5}{2}}\right)\,$	$=-{\tfrac {8}{15}}{\sqrt {\pi }}\,$	$\approx -0.945\,308\,720\,482\,941\,8812\,,$	A245887

General rational argument

In analogy with the half-integer formula,

{\begin{aligned}\Gamma \left(n+{\tfrac {1}{3}}\right)&=\Gamma \left({\tfrac {1}{3}}\right){\frac {(3n-2)!!!}{3^{n}}}\\\Gamma \left(n+{\tfrac {1}{4}}\right)&=\Gamma \left({\tfrac {1}{4}}\right){\frac {(4n-3)!!!!}{4^{n}}}\\\Gamma \left(n+{\tfrac {1}{p}}\right)&=\Gamma \left({\tfrac {1}{p}}\right){\frac {{\big (}pn-(p-1){\big )}!^{(p)}}{p^{n}}}\end{aligned}}

where $n! (p)$ denotes the $p$ th multifactorial of $n$ . Numerically,

\Gamma \left({\tfrac {1}{3}}\right)\approx 2.678\,938\,534\,707\,747\,6337

A073005

\Gamma \left({\tfrac {1}{4}}\right)\approx 3.625\,609\,908\,221\,908\,3119

A068466

\Gamma \left({\tfrac {1}{5}}\right)\approx 4.590\,843\,711\,998\,803\,0532

A175380

\Gamma \left({\tfrac {1}{6}}\right)\approx 5.566\,316\,001\,780\,235\,2043

A175379

\Gamma \left({\tfrac {1}{7}}\right)\approx 6.548\,062\,940\,247\,824\,4377

A220086

\Gamma \left({\tfrac {1}{8}}\right)\approx 7.533\,941\,598\,797\,611\,9047

A203142.

It is unknown whether these constants are transcendental in general, but $Γ(1 / 3)$ and $Γ(1 / 4)$ were shown to be transcendental by G. V. Chudnovsky. $Γ(1 / 4) / 4 \sqrt π$ has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that $Γ(1 / 4)$ , $π$ , and $e π$ are algebraically independent.

The number $Γ(1 / 4)$ is related to the lemniscate constant $S$ by

\Gamma \left({\tfrac {1}{4}}\right)={\sqrt {{\sqrt {2\pi }}S}},

and it has been conjectured by Gramain that

\Gamma \left({\tfrac {1}{4}}\right)={\sqrt[{4}]{4\pi ^{3}e^{2\gamma -\mathrm {\delta } +1}}}

where $δ$ is the Masser–Gramain constant A086058, although numerical work by Melquiond et al. indicates that this conjecture is false.^[1]

Borwein and Zucker have found that $Γ(n / 24)$ can be expressed algebraically in terms of $π$ , $K (k (1))$ , $K (k (2))$ , $K (k (3))$ , and $K (k (6))$ where $K (k (N))$ 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. No similar relations are known for $Γ(1 / 5)$ or other denominators.

In particular, $Γ(1 / 4)$ is given by

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

, where AGM() is the arithmetic–geometric mean.

and $Γ(1 / 6)$ is given by^[2]

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

Other formulas include the infinite products

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

and

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

where $A$ is the Glaisher-Kinkelin constant and $G$ is Catalan's constant.

C. H. Brown derived rapidly converging infinite series for particular values of the gamma function:^[3]

{\begin{aligned}{\frac {\left(\Gamma \left({\tfrac {1}{3}}\right)\right)^{6}}{12\pi ^{4}}}&={\frac {1}{\sqrt {10}}}\sum _{k=0}^{\infty }{\frac {(6k)!(-1)^{k}}{(k!)^{3}(3k)!3^{k}160^{3k}}}\\{\frac {\left(\Gamma \left({\tfrac {1}{4}}\right)\right)^{4}}{128\pi ^{3}}}&={\frac {1}{\sqrt {u}}}\sum _{k=0}^{\infty }{\frac {(6k)!(2w)^{k}}{(k!)^{3}(3k)!6486^{3k}}}\end{aligned}}

where,

{\begin{aligned}u&=273+180{\sqrt {2}}\\v&=1+{\sqrt {2}}\\w&=-761\,354\,780+538\,359\,129{\sqrt {2}}={\frac {6486^{3}}{2\left(uv^{2}{\sqrt {2}}\right)^{3}}}\end{aligned}}

equivalently,

{\frac {\left(\Gamma \left({\tfrac {1}{4}}\right)\right)^{4}}{128\pi ^{3}}}={\frac {1}{\sqrt {u}}}\sum _{k=0}^{\infty }{\frac {(6k)!}{(k!)^{3}(3k)!}}{\frac {1}{(uv^{2}{\sqrt {2}})^{3k}}}.

The following two representations for $Γ(3 / 4)$ were given by I. Mező^[4]

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

and

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

where $ϑ 1$ and $ϑ 4$ are two of the Jacobi theta functions.

Products

Some product identities include:

\prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}\approx 3.627\,598\,728\,468\,435\,7012

A186706

\prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}\approx 7.874\,804\,972\,861\,209\,8721

A220610

\prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}\approx 17.655\,285\,081\,493\,524\,2483

\prod _{r=1}^{5}\Gamma \left({\tfrac {r}{6}}\right)=4{\sqrt {\frac {\pi ^{5}}{3}}}\approx 40.399\,319\,122\,003\,790\,0785

\prod _{r=1}^{6}\Gamma \left({\tfrac {r}{7}}\right)={\frac {8\pi ^{3}}{\sqrt {7}}}\approx 93.754\,168\,203\,582\,503\,7970

\prod _{r=1}^{7}\Gamma \left({\tfrac {r}{8}}\right)=4{\sqrt {\pi ^{7}}}\approx 219.828\,778\,016\,957\,263\,6207

In general:

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

{\frac {\Gamma \left({\tfrac {1}{5}}\right)\Gamma \left({\tfrac {4}{15}}\right)}{\Gamma \left({\tfrac {1}{3}}\right)\Gamma \left({\tfrac {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}}}}}}}}}

^[5]

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

^[6]

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

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

Imaginary and complex arguments

The gamma function on the imaginary unit $i = \sqrt -1$ returns A212877, A212878:

\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i.

It may also be given in terms of the Barnes $G$ -function:

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

The gamma function with complex arguments returns

\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i

\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i

\Gamma ({\tfrac {1}{2}}+{\tfrac {1}{2}}i)\approx 0.818\,163\,9995-0.763\,313\,8287\,i

\Gamma ({\tfrac {1}{2}}-{\tfrac {1}{2}}i)\approx 0.818\,163\,9995+0.763\,313\,8287\,i

\Gamma (5+3i)\approx 0.016\,041\,8827-9.433\,293\,2898\,i

\Gamma (5-3i)\approx 0.016\,041\,8827+9.433\,293\,2897\,i.

Other constants

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

x_{\mathrm {min} }=1.461\,632\,144\,968\,362\,341\,262\ldots \,

A030169

with the value

\Gamma \left(x_{\mathrm {min} }\right)=0.885\,603\,194\,410\,888\ldots \,

A030171.

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 $Γ(x)$
$x$	$Γ(x)$	OEIS
3000495916991735544♠−0.5040830082644554092582693045	2999645535638884499♠−3.5446436111550050891219639933	A175472
2999842650152683761♠−1.5734984731623904587782860437	−7000230240725833968♠2.3024072583396801358235820396	A175473
2999738927913155585♠−2.6107208684441446500015377157	3000111863641598758♠−0.8881363584012419200955280294	A175474
2999636470663356309♠−3.6352933664369010978391815669	−6999245127539834366♠0.2451275398343662504382300889	A256681
2999534676223825685♠−4.6532377617431424417145981511	3001472203604126806♠−0.0527796395873194007604835708	A256682
2999433283755844311♠−5.6671624415568855358494741745	−6997932459448261485♠0.0093245944826148505217119238	A256683
2999332158178692657♠−6.6784182130734267428298558886	3002860260339105023♠−0.0013973966089497673013074887	A256684
2999231221167496837♠−7.6877883250316260374400988918	−6996181878444909404♠0.0001818784449094041881014174	A256685
2999130423583618359♠−8.6957641638164012664887761608	3004790747095534733♠−0.0000209252904465266687536973	A256686
2999029732745999813♠−9.7026725400018637360844267649	−6994215741610452285♠0.0000021574161045228505405031	A256687

The inverse of the gamma function gives out this interesting result :

$\int \limits _{1}^{\infty }{\frac {1}{\Gamma (x)}}dx\simeq 2.2665345076\dots$ also equivalent to $\int \limits _{0}^{\infty }{\frac {1}{\Gamma (x+1)}}dx=\int \limits _{0}^{\infty }{\frac {1}{x!}}dx\simeq 2.2665345076\dots$

References

↑ Melquiond, Guillaume; Nowak, W. Georg; Zimmermann, Paul (2013). "Numerical approximation of the Masser–Gramain constant to four decimal places". Math. Comp. 82: 1235–1246. doi:10.1090/S0025-5718-2012-02635-4.
↑ "Archived copy". Archived from the original on 2016-02-14. Retrieved 2015-03-09.
↑ Cetin Hakimgolu-Brown : iamned.com math page Archived October 9, 2016, at the Wayback Machine.
↑ Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
↑ Raimundas Vidūnas, Expressions for Values of the Gamma Function
↑ Weisstein, Eric W. "Gamma Function". MathWorld.

Gramain, F. (1981). "Sur le théorème de Fukagawa-Gel'fond". Invent. Math. 63 (3): 495–506. doi:10.1007/BF01389066.
Borwein, J. M.; Zucker, I. J. (1992). "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind". IMA J. Numerical Analysis. 12 (4): 519–526. MR 1186733. doi:10.1093/imanum/12.4.519.
X. Gourdon & P. Sebah. Introduction to the Gamma Function
S. Finch. Euler Gamma Function Constants
Weisstein, Eric W. "Gamma Function". MathWorld.

Vidunas, Raimundas. "Expressions for values of the gamma function". arXiv:math.CA/0403510 .
Vidunas, Raimundas (2005). "Expressions for values of the gamma function". Kyushu J. Math. 59 (2): 267–283. MR 2188592. doi:10.2206/kyushujm.59.267.
Adamchik, V. S. (2005). "Multiple Gamma Function and Its Application to Computation of Series" (PDF). The Ramanujan Journal. 9 (3): 271–288. MR 2173489. doi:10.1007/s11139-005-1868-3.
Duke, W.; Imamoglu, Ö. (2006). "Special values of multiple gamma functions" (PDF). J. Theor. Nombres Bordeaux. 18 (1): 113–123. MR 2245878. doi:10.5802/jtnb.536.

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