Lemniscate elliptic functions: Difference between revisions

Revision as of 10:13, 1 July 2025

Revision as of 05:41, 2 July 2025

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:<math> z = \int_0^{\operatorname{sl} z}\frac{\mathrm{d}t}{\sqrt{1-t^4}} = \int_{\operatorname{cl} z}^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}.</math>

Beyond that square, the functions can be [[analytic continuation|analytically continued]]extended to the whole [[complex plane]] byvia [[analytic continuation]] aby series ofsuccessive [[Schwarz reflection principle|reflections]].

By comparison, the circular sine and cosine can be defined as the solution to the initial value problem: