Particular values of the gamma function: Difference between revisions

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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]].

The number {{math|Γ({{sfrac|1|4}})}} is related to the [[Gauss'slemniscate constant]] {{mvar|Gϖ}} by

:<math>\Gamma\left(\tfrac14\right) = \sqrt{2G2\varpi\sqrt{2\pi^3}},</math>

and it has been conjectured by Gramain that