Lemniscate elliptic functions: Difference between revisions

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[[File:Slh in the complex plane.png|thumb|The hyperbolic lemniscate sine in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.]]

For convenience, let <math>\sigma=\sqrt{2}\varpi</math>. <math>\sigma</math> is the "squircular" analog of <math>\pi</math> (see below). The decimal expansion of <math>\sigma</math> (i.e. <math>3.7081\ldots</math><ref>http://oeis.org/A175576 {{Barecite URL inlineOEIS|date=August 2024A175576}}</ref>) appears in entry 34e of chapter 11 of Ramanujan's second notebook.<ref>{{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part II |publisher=Springer |year=1989 |isbn=978-1-4612-4530-8}} p. 96</ref>

The hyperbolic lemniscate sine ({{math|slh}}) and cosine ({{math|clh}}) can be defined as inverses of elliptic integrals as follows: