Particular values of the gamma function: Difference between revisions

Revision as of 05:10, 9 August 2024

Revision as of 11:53, 20 August 2024

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where <math>\gamma</math> is the [[Euler–Mascheroni constant]] and <math>\sim</math> denotes [[Asymptotic analysis|asymptotic equivalence]].

It is unknown whether these constants are [[transcendental number|transcendental]] in general, but {{<math|Γ>\Gamma\left({{sfrac|1|3}}\tfrac13\right)}}</math> and {{<math|Γ>\Gamma\left({{sfrac|1|4}}\tfrac14\right)}}</math> were shown to be transcendental by [[Chudnovsky brothers|G. V. Chudnovsky]]. {{<math|Γ(>\pi^{{sfrac|1|4}-\frac14}\Gamma\left(\tfrac14\right) <big><big>/</big></bigmath> {{radic|π|4}}}} has also long been known to be transcendental, and [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved in 1996 that {{<math|Γ>\Gamma\left({{sfrac|1|4}}\tfrac14\right)}}, {{\pi</math|π}},> and {{<math|''>e''<sup>π^\pi</supmath>}} are [[algebraically independent]].

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

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

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|s2cid=123079859

}}</ref> that

<!-- :<math>\log \Gamma(1/4) = \frac{1}{4}(1 + 3 \log \pi + 2 \log 2 + 2 \gamma - \mathrm{\rho})</math> --><math>\Gamma \left (\tfrac14 \right ) = \sqrt[4]{4 \pi^3 e^{2 \gamma -\mathrm{\delta}+1}}</math>

:<math>\Gamma \left (\tfrac14 \right ) = \sqrt[4]{4 \pi^3 e^{2 \gamma -\mathrm{\delta}+1}}</math>

where {{mvar|δ}} is the [[Masser–Gramain constant]] {{OEIS2C|A086058}}, although numerical work by Melquiond et al. indicates that this conjecture is false.<ref>{{cite journal|doi=10.1090/S0025-5718-2012-02635-4 |first1=Guillaume|last1= Melquiond|first2=W. Georg |last2=Nowak|first3=Paul |last3=Zimmermann|journal=Math. Comp.|title=Numerical approximation of the Masser–Gramain constant to four decimal places|year=2013|volume=82|issue=282|pages=1235–1246|doi-access=free}}</ref>

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|mr=1186733

|doi=10.1093/imanum/12.4.519

}}</ref> have found that {{<math|Γ>\Gamma\left(\tfrac{n}{sfrac|''n''|24}}\right)}}</math> can be expressed algebraically in terms of {{mvar|π}}, {{math|''K''(''k''(1))}}, {{math|''K''(''k''(2))}}, {{math|''K''(''k''(3))}}, and {{math|''K''(''k''(6))}} where {{math|''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 [[quadratic convergence|quadratically convergent]] [[arithmetic–geometric mean]] iterations. For example:

:<math>\begin{align}

\Gamma \left(\tfrac16 \right) &= \frac{\sqrt{\frac{3}{\pi }} \Gamma\left(\frac{1}{3}\right)^2}{\sqrt[3]{2}} \\

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where {{mvar|A}} is the [[Glaisher–Kinkelin constant]] and {{mvar|G}} is [[Catalan's constant]].

The following two representations for {{<math|Γ>\Gamma\left({{sfrac|3|4}}\tfrac34\right)}}</math> were given by I. Mező<ref name="Mezo2">{{Citation

| last = Mező

| first = István

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:<math>\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}}</math>

and many more relations for {{<math|Γ>\Gamma\left(\tfrac{n}{sfrac|''n''|''d''}}\right)}}</math> where the denominator d divides 24 or 60.<ref>{{cite journal

| last = Vidūnas | first = Raimundas

| arxiv = math/0403510

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:<math>\Gamma(i) = \frac{G(1+i)}{G(i)} = e^{-\log G(i)+ \log G(1+i)}.</math>

Because of the [[Reflection formula|Euler Reflection Formula]], and the fact that <math>\Gamma(\baroverline{z})=\baroverline{\Gamma}(z)}</math>, we have an expression for the [[modulus squared]] of the gamma function evaluated on the imaginary axis:

:<math>\left|\Gamma(i\kappa)\right|^2=\frac{\pi}{\kappa\sinh(\pi\kappa)}</math>