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m one link to 'pendulum' suffices here |
Explain why it's well-defined |
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:<math>\mathcal{E}(u,m)=E(\operatorname{am}(u,m),m)</math>
for <math>u\in\mathbb{R}</math> and <math>0<m<1</math> and by [[analytic continuation]] in each of the variables otherwise: the Jacobi epsilon function is meromorphic in the whole complex plane (in both <math>u</math> and <math>m</math>). Alternatively, throughout both the <math>u</math>-plane and <math>m</math>-plane,<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=22.16.E17|title=Jacobian Elliptic Functions}}</ref>
:<math>\mathcal{E} (u,m)=\int_0^u \operatorname{dn}^2(t,m)\, \mathrm dt
<math>\mathcal{E}</math> is well-defined in this way because all [[Residue (complex analysis)|residues]] of <math>t\mapsto\operatorname{dn}(t,m)^2</math> are zero, so the integral is path-independent. So the Jacobi epsilon relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind:
:<math>E(\varphi,m)=\mathcal{E}(F(\varphi,m),m).</math>
The Jacobi epsilon function is not an elliptic function, but it appears when differentiating the Jacobi elliptic functions with respect to the parameter.
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:<math>\sin(\operatorname{am}(u+v)+\operatorname{am}(u-v))=\frac{2\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v}{1-m\operatorname{sn}^2u\operatorname{sn}^2v},</math>
:<math>\cos(\operatorname{am}(u+v)-\operatorname{am}(u-v))=\dfrac{\operatorname{cn}^2v-\operatorname{sn}^2v\operatorname{dn}^2u}{1-m\operatorname{sn}^2u\operatorname{sn}^2v}
where both identities are valid for all <math>u,v,m\in\mathbb{C}</math> such that both sides are well-defined.
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