Particular values of the Gamma function

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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[edit]

For positive integer arguments, the Gamma function coincides with the factorial, that is,

\Gamma(n+1) = n!\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\,, OEISA002161
\Gamma(\tfrac32)\, = \frac {1}{2} \sqrt{\pi}\, \approx 0.8862269254527580137\,, OEISA019704
\Gamma(\tfrac52)\, = \frac {3}{4} \sqrt{\pi}\, \approx 1.3293403881791370205\,,
\Gamma(\tfrac72)\, = \frac {15}{8} \sqrt{\pi}\, \approx 3.3233509704478425512\,,

and by means of the reflection formula,

\Gamma(-\tfrac12)\, = -2\sqrt{\pi}\, \approx -3.5449077018110320546\,, OEISA019707
\Gamma(-\tfrac32)\, = \frac {4}{3} \sqrt{\pi}\, \approx 2.3632718012073547031\,,
\Gamma(-\tfrac52)\, = -\frac {8}{15} \sqrt{\pi}\, \approx -0.9453087204829418812\,.

General rational arguments[edit]

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 OEISA073005
\Gamma(\tfrac14) \approx 3.6256099082219083119 OEISA068466
\Gamma(\tfrac15) \approx 4.5908437119988030532 OEISA175380
\Gamma(\tfrac16) \approx 5.5663160017802352043 OEISA175379
\Gamma(\tfrac17) \approx 6.5480629402478244377 OEISA220086
\Gamma(\tfrac18) \approx 7.5339415987976119047 OEISA203142.

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 OEISA086058.

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)}.

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:[1]

\begin{align}
\frac{\left(\Gamma(\tfrac13) \right )^6\sqrt{10}}{12\pi^4} &= \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}\\
w &= 538359129\sqrt{2}-761354780
\end{align}

or, since 2w/6486^3 in fact is a cube involving u,

\frac{[\Gamma(\tfrac14)]^4}{128\pi^3} = \frac{1}{\sqrt{u}} \sum_{k = 0}^{\infty} \frac{(6k)!}{(k!)^{3}(3k)!} \frac{1}{(u\sqrt{2}(1+\sqrt{2})^2)^{3k}}.

Products[edit]

Some product identities include:

 \prod_{r=1}^2 \Gamma(\tfrac{r}{3}) = \frac{2\pi}{\sqrt{3}} \approx 3.6275987284684357012 OEISA186706
 \prod_{r=1}^3 \Gamma(\tfrac{r}{4}) = \sqrt{2\pi^3} \approx 7.8748049728612098721 OEISA220610
 \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
\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}}}}}[2]
\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}[3]

Imaginary unit[edit]

The gamma function on the imaginary unit i = \sqrt{-1} returns OEISA212877, OEISA212878:

\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[edit]

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[edit]

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

x_\mathrm{min} = 1.461632144968362341262\ldots\, OEISA030169

with the value

\Gamma(x_\mathrm{min}) = 0.885603194410888\ldots\, OEISA030171.

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
-0.5040830082644554092582693045 -3.5446436111550050891219639933 OEISA175472
-1.5734984731623904587782860437 2.3024072583396801358235820396 OEISA175473
-2.6107208684441446500015377157 -0.8881363584012419200955280294 OEISA175474
-3.6352933664369010978391815669 0.2451275398343662504382300889
-4.6532377617431424417145981511 -0.0527796395873194007604835708
-5.6671624415568855358494741745 0.0093245944826148505217119238
-6.6784182130734267428298558886 -0.0013973966089497673013074887
-7.6877883250316260374400988918 0.0001818784449094041881014174
-8.6957641638164012664887761608 -0.0000209252904465266687536973
-9.7026725400018637360844267649 0.0000021574161045228505405031

See also[edit]

References[edit]

  1. ^ Cetin Hakimgolu-Brown : iamned.com math page
  2. ^ Raimundas Vidūnas, Expessions for Values of the Gamma Function
  3. ^ Weisstein, Eric W., "Gamma Function", MathWorld.