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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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>

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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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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}}}}.<ref>.. {{OEIS2C|A218802}}</ref>.

==See also==