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→{{anchor|sn|cn|dn|am}}Definition in terms of inverses of elliptic integrals: Unify a characterization of epsilon with that of sn, cn, dn |
→Jacobi elliptic functions as solutions of nonlinear ordinary differential equations: Derivatives with respect to the second variable |
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==Jacobi elliptic functions as solutions of nonlinear ordinary differential equations==
===Derivatives with respect to the first variable===
The [[derivative]]s of the three basic Jacobi elliptic functions (with respect to the first variable, with <math>m</math> fixed) are:
<math display=block>\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{sn}(z) = \operatorname{cn}(z) \operatorname{dn}(z),</math>
<math display=block>\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{cn}(z) = -\operatorname{sn}(z) \operatorname{dn}(z),</math>
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:<math>\theta=2\arcsin (k\operatorname{cd}(\sqrt{c}t,k^2)),\quad k=\sin\frac{\theta_0}{2}.</math>
===Derivatives with respect to the second variable===
With the first argument <math>z</math> fixed, the derivatives with respect to the second variable <math>m</math> are as follows:
:<math>\begin{align}\frac{\mathrm d}{\mathrm dm}\operatorname{sn}(z)&=\frac{\operatorname{dn}(z)\operatorname{cn}(z)((1-m)z-\mathcal{E}(z)+m\operatorname{cd}(z)\operatorname{sn}(z))}{2m(1-m)},\\
\frac{\mathrm d}{\mathrm dm}\operatorname{cn}(z)&=\frac{\operatorname{sn}(z)\operatorname{dn}(z)((m-1)z+\mathcal{E}(z)-m\operatorname{sn}(z)\operatorname{cd}(z))}{2m(1-m)},\\
\frac{\mathrm d}{\mathrm dm}\operatorname{dn}(z)&=\frac{\operatorname{sn}(z)\operatorname{cn}(z)((m-1)z+\mathcal{E}(z)-\operatorname{dn}(z)\operatorname{sc}(z))}{2(1-m)}.\end{align}</math>
==Expansion in terms of the nome==
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