Editing Elliptic integral (section)

===Relation to the gamma function===
If {{math|1=''k''<sup>2</sup> = ''λ''(''i''{{sqrt|''r''}})}} and <math>r \isin \mathbb{Q}^+</math> (where {{mvar|λ}} is the [[modular lambda function]]), then {{math|''K''(''k'')}} is expressible in closed form in terms of the [[gamma function]].<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 296</ref> For example, {{math|1=''r'' = 2}}, {{math|1=''r'' = 3}} and {{math|1=''r'' = 7}} give, respectively,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 298</ref>

<math display="block">K\left(\sqrt{2}-1\right)=\frac{\Gamma \left(\frac18\right)\Gamma \left(\frac38\right)\sqrt{\sqrt{2}+1}}{8\sqrt[4]{2}\sqrt{\pi}},</math>

and

<math display="block">K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)=\frac{1}{8\pi}\sqrt[4]{3}\,\sqrt[3]{4}\,\Gamma\biggl(\frac{1}{3}\biggr)^3</math>

and

<math display="block">K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)=\frac{\Gamma \left(\frac17\right)\Gamma \left(\frac27\right)\Gamma \left(\frac47\right)}{4\sqrt[4]{7}\pi}.</math>

More generally, the condition that
<math display="block">\frac{iK'}{K}=\frac{iK\left(\sqrt{1-k^2}\right)}{K(k)}</math>
be in an [[Quadratic field|imaginary quadratic field]]<ref group="note">{{mvar|K}} can be [[Analytic continuation|analytically extended]] to the [[complex plane]].</ref> is sufficient.<ref>{{Cite journal|title=On Epstein's Zeta Function (I).|last1=Chowla|first1=S.|last2=Selberg|first2=A.|journal=Proceedings of the National Academy of Sciences|year=1949|volume=35|issue=7|page=373|doi=10.1073/PNAS.35.7.371|pmid=16588908|pmc=1063041|bibcode=1949PNAS...35..371C|s2cid=45071481}}</ref><ref>{{Cite journal|url=https://eudml.org/doc/150803|title=On Epstein's Zeta-Function|last1=Chowla|first1=S.|last2=Selberg|first2=A.|journal=Journal für die Reine und Angewandte Mathematik|year=1967|volume=227|pages=86–110}}</ref> For instance, if {{math|1=''k'' = ''e''<sup>5''πi''/6</sup>}}, then {{math|1={{sfrac|''iK''{{prime}}|''K''}} = ''e''<sup>2''πi''/3</sup>}} and<ref>{{Cite web|url=https://fungrim.org/topic/Legendre_elliptic_integrals/|title = Legendre elliptic integrals (Entry 175b7a)}}</ref>

<math display="block">K\left(e^{5\pi i/6}\right)=\frac{e^{-\pi i/12}\Gamma ^3\left(\frac13\right)\sqrt[4]{3}}{4\sqrt[3]{2}\pi}.</math>