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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>\
:<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>
where <math>\mathbb{H}</math> is the upper half-plane in the complex plane, <math>\partial F_1</math> is the boundary of <math>F_1</math> and
:<math>F_1=\{\tau\in\mathbb{H}:\left|\operatorname{Re}\tau\right|\le 1,\left|\operatorname{Re}(1/\tau)\right|\le 1\}.</math>
In this way, each <math>m\,\overset{\text{def}}{=}\,
:<math>\begin{align}\operatorname{sn}(u,m)&=\frac{\theta_3(\tau)\theta_1(\zeta |\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
\operatorname{cn}(u,m)&=\frac{\theta_4(\tau)\theta_2(\zeta |\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
\operatorname{dn}(u,m)&=\frac{\theta_4(\tau)\theta_3(\zeta |\tau)}{\theta_3(\tau)\theta_4(\zeta|\tau)}\end{align}</math>
where <math>\zeta=u/\theta_3(\tau)^2</math>.
In the book, they place an additional restriction on <math>m</math> (that <math>m\notin (-\infty,0)\cup (1,\infty)</math>), but it is in fact not a necessary restriction (see the Cox reference). Also, if <math>m=0</math> or <math>m=1</math>, the Jacobi elliptic functions degenerate to non-elliptic functions which is described below.
==Definition in terms of Neville theta functions==
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