Particular values of the gamma function: Difference between revisions

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{{Short description|Mathematical constants}}

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

==Integers and half-integers==

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

For positive half-integers <math> \frac{k}{2} </math> where <math> k\in 2\mathbb{N}^*+1 </math> is an odd integer greater or equal <math>3</math>, the function values are given exactly by

:<math>\Gamma \left (\tfrac{nk}{2} \right) = \sqrt \pi \frac{(nk-2)!!}{2^\frac{nk-1}{2}}\,,</math>

or equivalently, for non-negative integer values of&nbsp;{{mvar|n}}:

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It is unknown whether these constants are [[transcendental number|transcendental]] in general, but {{math|Γ({{sfrac|1|3}})}} and {{math|Γ({{sfrac|1|4}})}} were shown to be transcendental by [[Chudnovsky brothers|G. V. Chudnovsky]]. {{math|Γ({{sfrac|1|4}}) <big><big>/</big></big> {{radic|π|4}}}} has also long been known to be transcendental, and [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved in 1996 that {{math|Γ({{sfrac|1|4}})}}, {{math|π}}, and {{math|''e''^π}} are [[algebraically independent]].

For <math>n\geq 2</math> at least one of the two numbers  {{<math|Γ>\Gamma\left({{sfrac|1|n}}\tfrac1n\right)}}</math> and {{<math|Γ>\Gamma\left({{sfrac|2|n}}\tfrac2n\right)}}</math> is transcendental.<ref>{{Cite journal |last=Waldschmidt |first=Michel |date=2006 |title=Transcendence of periods: the state of the art |url=https://hal.science/hal-00411301 |journal=Pure and Applied Mathematics Quarterly |volume=2 |issue=2 |pages=435–463|doi=10.4310/PAMQ.2006.v2.n2.a3 }}</ref>

The number {{<math|Γ>\Gamma\left({{sfrac|1|4}}\tfrac14\right)}}</math> is related to the [[lemniscate constant]] {{mvar|<math>\varpi</math>}} by

:<math>\Gamma\left(\tfrac14\right) = \sqrt{2\varpi\sqrt{2\pi}}</math>

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\Gamma \left(\tfrac14 \right) &= 2\sqrt{K\left( \tfrac 12 \right)\sqrt{\pi}} \\

\Gamma \left(\tfrac13 \right) &= \frac{2^{7/9} \sqrt[3]{\pi K\left(\frac{1}{4}\left(2-\sqrt{3}\right)\right)}}{\sqrt[12]{3}} \\

\Gamma \left(\fractfrac{1}{8}\right) \Gamma \left(\fractfrac{3}{8}\right) &= 8 \sqrt[4]{2} \sqrt{\left(\sqrt{2}-1\right) \pi } K\left(3-2 \sqrt{2}\right) \\

\frac{\Gamma \left(\frac{1}{8}\right)}{\Gamma \left(\frac{3}{8}\right)} &= \frac{2 \sqrt{\left(1+\sqrt{2}\right) K\left(\frac{1}{2}\right)}}{\sqrt[4]{\pi }}

\end{align}</math>

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No similar relations are known for {{math|Γ({{sfrac|1|5}})}} or other denominators.

In particular, where AGM() is the [[arithmetic–geometric mean]], we have<ref>{{cite web|url=https://math.stackexchange.com/q/1631760 |title=Archived copy |accessdate=2015-03-09 |url-status=live }}</ref>

:<math>\Gamma\left(\tfrac13\right) = \frac{2^\frac{7}{9}\cdot \pi^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13}</math>

:<math>\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)}</math>

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where {{math|''θ''₁}} and {{math|''θ''₄}} are two of the [[Theta function|Jacobi theta functions]].

There also exist a number of [[Carl Johan Malmsten|Malmsten integrals]] for certain values of the gamma function:<ref name=":1">{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303|url-access=subscription }}</ref>

:<math>\int_1^\infty \frac{\ln \ln t}{1+t^2} = \frac\pi4\left(2\ln2 + 3\ln\pi-4\Gamma\left(\tfrac14\right)\right)</math>

:<math>\int_1^\infty \frac{\ln \ln t}{1+t+t^2} = \frac\pi{6\sqrt3}\left(8\ln2 -3\ln3 + 8\ln\pi -12\Gamma\left(\tfrac13\right)\right)</math>

== Products ==

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

:<math>x_{\min} = 1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots\,</math> {{OEIS2C|A030169}}

with the value

:<math>\Gamma\left(x_{\min}\right) = 0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots\,</math> {{OEIS2C|A030171}}.

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

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| {{val|−9.7026725400018637360844267649}} || {{0|−}}{{val|0.0000021574161045228505405031}} || {{OEIS2C|A256687}}

|}

The only values of {{math|''x'' > 0}} for which {{math|1=Γ(''x'') = ''x''}} are {{math|1=''x'' = 1}} and {{math|''x'' ≈ {{val|3.5623822853908976914156443427}}}}... {{OEIS2C|A218802}}.

==See also==

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==References==

<references />

==Further reading==

* {{Cite journal

|first1=F.

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|bibcode=1981InMat..63..495G

|s2cid=123079859

|url=https://www.semanticscholar.org/paper/8483977e31096fef85a31b0aaa24a16e602cc42f

}}

* {{Cite journal

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}}

* X. Gourdon & P. Sebah. [http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html Introduction to the Gamma Function]

* S. Finch. [http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Number_Theory_Constants/2002.asp Euler Gamma Function Constants]{{dead link|date=May 2010}}

* {{MathWorld|title=Gamma Function|urlname=GammaFunction}}

* {{cite journal