Jacobi elliptic functions: Difference between revisions

Revision as of 21:27, 27 August 2024

Revision as of 21:39, 27 August 2024

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===Using modular inversion===

In fact, the definition of the Jacobi elliptic functions in Whittaker & Watson is stated a little bit differently than the one given above (but it's equivalent to it) and relies on modular inversion: The function <math>\nu</math>, defined by

:<math>\nu (t\tau)=\frac{\theta_4(\tau)^4}{\theta_3(\tau)^4},</math>

assumes every value in <math>\mathbb{C}-\{0,1\}</math> ''once and only once''<ref>{{cite journal |last=Cox |first=David Archibald |authorlink1=David A. Cox |date=January 1984 |title=The Arithmetic-Geometric Mean of Gauss|url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique|volume=30|issue=2|pages=290}}</ref> in

:<math>F_1-(\partial F_1\cap\{\tau\in\mathbb{H}:\operatorname{Re}\tau <0\})</math>