Particular values of the gamma function: Difference between revisions

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

and it has been conjectured by Gramain<ref>{{Cite thatjournal

|first1=F.

|last1=Gramain

|title=Sur le théorème de Fukagawa-Gel'fond

|journal=Invent. Math.

|volume=63

|number=3

|doi=10.1007/BF01389066

|year=1981

|pages=495&ndash;506

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

|s2cid=123079859

}}</ref> that

<!-- :<math>\log \Gamma(1/4) = \frac{1}{4}(1 + 3 \log \pi + 2 \log 2 + 2 \gamma - \mathrm{\rho})</math> -->

Line 113 ⟶ 125:

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>

*Borwein and Zucker<ref>{{Cite journal

Borwein and Zucker have found that {{math|Γ({{sfrac|''n''|24}})}} 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:

|first1=J. M.

|last1=Borwein

|first2=I. J.

|last2=Zucker

|title=Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind

|journal=IMA Journal of Numerical Analysis

|volume=12

|issue=4

|pages=519&ndash;526

|year=1992

|mr=1186733

|doi=10.1093/imanum/12.4.519

Borwein and Zucker}}</ref> have found that {{math|Γ({{sfrac|''n''|24}})}} 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}} \\

Line 257 ⟶ 282:

==References==

{{reflist}}

<references />

* {{Cite journal

|first1=F.

|last1=Gramain

|title=Sur le théorème de Fukagawa-Gel'fond

|journal=Invent. Math.

|volume=63

|number=3

|doi=10.1007/BF01389066

|year=1981

|pages=495&ndash;506

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

|s2cid=123079859

}}

==Further reading==

* {{Cite journal

|first1=J. M.

|last1=Borwein

|first2=I. J.

|last2=Zucker

|title=Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind

|journal=IMA Journal of Numerical Analysis

|volume=12

|issue=4

|pages=519&ndash;526

|year=1992

|mr=1186733

|doi=10.1093/imanum/12.4.519

}}

* {{Cite journal

|first1=V. S.