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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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-Integral 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’sMalmsten'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(\frac14tfrac14\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(\frac13tfrac13\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==