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→Jacobi elliptic functions as solutions of nonlinear ordinary differential equations: Derivatives with respect to the second variable |
→{{anchor|sn|cn|dn|am}}Definition in terms of inverses of elliptic integrals: It's better not to introduce it if we don't use it |
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However, the integral inversion above defines a unique single-valued real-analytic function in a real [[Neighbourhood (mathematics)|neighborhood]] of <math>u=0</math> if <math>m</math> is real. There is a unique [[analytic continuation]] of this function from that neighborhood to <math>u\in\mathbb{R}</math>. The analytic continuation of this function is periodic in <math>u</math> if and only if <math>m>1</math> (with the minimal period <math>4K(1/m)/\sqrt{m}</math>), and it is denoted by <math>\operatorname{am}(u,m)</math> in the rest of this article.
The '''Jacobi epsilon''' function can be defined as
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