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^{(n-1)/2}}\,,

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

\begin{align} \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{align}

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

\Gamma(\tfrac12)\, = \sqrt{\pi}\, \approx 1.7724538509055160273\,,  A002161
\Gamma(\tfrac32)\, = \frac {1}{2} \sqrt{\pi}\, \approx 0.8862269254527580137\,,  A019704
\Gamma(\tfrac52)\, = \frac {3}{4} \sqrt{\pi}\, \approx 1.3293403881791370205\,,  A245884
\Gamma(\tfrac72)\, = \frac {15}{8} \sqrt{\pi}\, \approx 3.3233509704478425512\,,  A245885

and by means of the reflection formula,

\Gamma(-\tfrac12)\, = -2\sqrt{\pi}\, \approx -3.5449077018110320546\,,  A019707
\Gamma(-\tfrac32)\, = \frac {4}{3} \sqrt{\pi}\, \approx 2.3632718012073547031\,,  A245886
\Gamma(-\tfrac52)\, = -\frac {8}{15} \sqrt{\pi}\, \approx -0.9453087204829418812\,,  A245887

General rational arguments

In analogy with the half-integer formula,

\begin{align} \Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac{(3n-2)!^{(3)}}{3^n} \\ \Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac{(4n-3)!^{(4)}}{4^n} \\ \Gamma \left(n+\tfrac{1}{p} \right ) &= \Gamma \left(\tfrac{1}{p} \right ) \frac{(pn-(p-1))!^{(p)}}{p^n} \end{align}

where n!^{(k)} denotes the k:th multifactorial of n. Numerically,

\Gamma(\tfrac13) \approx 2.6789385347077476337  A073005
\Gamma(\tfrac14) \approx 3.6256099082219083119  A068466
\Gamma(\tfrac15) \approx 4.5908437119988030532  A175380
\Gamma(\tfrac16) \approx 5.5663160017802352043  A175379
\Gamma(\tfrac17) \approx 6.5480629402478244377  A220086
\Gamma(\tfrac18) \approx 7.5339415987976119047  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. \Gamma(\tfrac14) / \pi^{1/4} has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and eπ are algebraically independent.

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

\Gamma(\tfrac14) = \sqrt{\sqrt{2 \pi} S},

and it has been conjectured by Gramain that

\Gamma \left (\tfrac14 \right ) = \left(4 \pi^3 e^{2 \gamma -\mathrm{\delta}+1}\right)^{\frac14}

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(\tfrac14) = \sqrt \frac{(2 \pi)^{\frac{3}{2}}}{AGM(\sqrt 2, 1)}

and Γ(1/6) is given by

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

Other formulas include the infinite products

\Gamma(\tfrac14) = (2 \pi)^{\frac{3}{4}} \prod_{k=1}^\infty \tanh \left( \frac{\pi k}{2} \right)

and

\Gamma(\tfrac14) = 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:[2]

\begin{align} \frac{\left(\Gamma(\tfrac13) \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(\tfrac14) \right )^4}{128\pi^3} &= \frac{1}{\sqrt{u}}\, \sum_{k = 0}^{\infty} \frac{(6k)!(2w)^k}{(k!)^{3}(3k)! 6486^{3k}} \end{align}

where,

\begin{align} u &= 273+180\sqrt{2}\\ v &= 1+\sqrt{2}\\ w &= -761354780+538359129\sqrt{2} = \frac{6486^3}{2\bigl(uv^2\sqrt{2}\bigr)^3} \end{align}

equivalently,

\frac{\left(\Gamma(\tfrac14) \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

[3]
\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\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(\frac34\right)}=\sum_{k=-\infty}^\infty\frac{\vartheta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}},

where \vartheta_1 and \vartheta_4 are two of the Jacobi theta functions.

Products

Some product identities include:

\prod_{r=1}^2 \Gamma(\tfrac{r}{3}) = \frac{2\pi}{\sqrt{3}} \approx 3.6275987284684357012  A186706
\prod_{r=1}^3 \Gamma(\tfrac{r}{4}) = \sqrt{2\pi^3} \approx 7.8748049728612098721  A220610
\prod_{r=1}^4 \Gamma(\tfrac{r}{5}) = \frac{4\pi^2}{\sqrt{5}} \approx 17.6552850814935242483
\prod_{r=1}^5 \Gamma(\tfrac{r}{6}) = 4\sqrt{\frac{\pi^5}{3}} \approx 40.3993191220037900785
\prod_{r=1}^6 \Gamma(\tfrac{r}{7}) = \frac{8\pi^3}{\sqrt{7}} \approx 93.7541682035825037970
\prod_{r=1}^7 \Gamma(\tfrac{r}{8}) = 4\sqrt{\pi^7} \approx 219.8287780169572636207

In general:

\prod_{r=1}^n \Gamma(\tfrac{r}{n+1}) = \sqrt{\frac{(2\pi)^n}{n+1}}
\frac{\Gamma(\tfrac{1}{5})\Gamma(\tfrac{4}{15})}{\Gamma(\tfrac{1}{3})\Gamma(\tfrac{2}{15})} = \frac{\sqrt{2}\sqrt[20]{3}}{\sqrt[6]{5} \sqrt[4]{5-\frac{7}{\sqrt{5}}+\sqrt{6-\frac{6}{\sqrt{5}}}}}[4]
\frac{\Gamma(\tfrac{1}{20})\Gamma(\tfrac{9}{20})}{\Gamma(\tfrac{3}{20})\Gamma(\tfrac{7}{20})} = \frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2}[5]

Imaginary unit

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)}.

Complex Arguments

The gamma function with the complex Arguments i = \sqrt{-1} returns

\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i
\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i
\Gamma(0.5 + 0.5i) \approx 0.8181639995 - 0.7633138287 i
\Gamma(0.5 - 0.5i) \approx 0.8181639995 + 0.7633138287 i
\Gamma(5 + 3i) \approx 0.0160418827 - 9.4332932898 i
\Gamma(5 - 3i) \approx 0.0160418827 + 9.4332932897 i.

Other constants

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

x_\mathrm{min} = 1.461632144968362341262\ldots\,  A030169

with the value

\Gamma(x_\mathrm{min}) = 0.885603194410888\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)
xΓ(x)OEIS
−0.5040830082644554092582693045 −3.5446436111550050891219639933  A175472
−1.5734984731623904587782860437 2.3024072583396801358235820396  A175473
−2.6107208684441446500015377157 −0.8881363584012419200955280294  A175474
−3.6352933664369010978391815669 0.2451275398343662504382300889  A256681
−4.6532377617431424417145981511 −0.0527796395873194007604835708  A256682
−5.6671624415568855358494741745 0.0093245944826148505217119238  A256683
−6.6784182130734267428298558886 −0.0013973966089497673013074887  A256684
−7.6877883250316260374400988918 0.0001818784449094041881014174  A256685
−8.6957641638164012664887761608 −0.0000209252904465266687536973  A256686
−9.7026725400018637360844267649 0.0000021574161045228505405031  A256687

See also

References

  1. ↑ Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann : Numerical approximation of the Masser-Gramain constant to four decimal places
  2. ↑ Cetin Hakimgolu-Brown : iamned.com math page
  3. ↑ 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
  4. ↑ Raimundas Vidūnas, Expessions for Values of the Gamma Function
  5. ↑ Weisstein, Eric W., "Gamma Function", MathWorld.
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