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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: [[Modular lambda function|The function]] <math>\lambda</math>, defined by
[[File:The region F1 for modular inversion.jpg|thumb|The region <math>F_1</math> in the complex plane. It is bounded by two semicircles from below, by a ray from the left and by a ray from the right.]]
:<math>\lambda (\tau)=\frac{\theta_2(\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
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