Reference: Elliptic Functions
Doubly-periodic functions on the complex plane. Here's a high-level overview.
(Source: Mostly Wikipedia. But if this reference is insufficient, try this one.)
Jacobi
16 functions: \(pq(u,m) = pq(u;k)\) (with \(m = k^2\)), as \(p\), \(q\) range over \(\{c,s,d,n\}\). Related by the equations \(pp = 1\) and \(pq \cdot qr = pr\).
Also, Jacobi amplitude \(\mathrm{am}(u;k)\), Jacobi epsilon \(\mathcal{E}(u;k)\), and elliptic integrals: \[\begin{align*} F(x;k) &= \int_0^x \sqrt{\frac{1}{(1-t^2)(1-k^2t^2)}} \mathrm{d}t &\qquad K(k) &= F(1;k) \\ E(x;k) &= \int_0^x \sqrt{\frac{1-k^2t^2}{1-t^2}} \mathrm{d}t &\qquad E(k) &= E(1;k) \end{align*}\]
Useful properties:
\[\begin{align*} F &= \mathrm{sn}^{-1} \\ E(x;k) &= \mathcal{E}(F(x;k);k) \end{align*}\]
\[\begin{align*} \mathrm{sn}(u;k) &= \sin\mathrm{am}(u;k) \\ \mathrm{cn}(u;k) &= \cos\mathrm{am}(u;k) \\ \mathrm{dn}(u;k) &= \tfrac{\mathrm{d}}{\mathrm{d}u} \mathrm{am}(u;k) \\ \mathrm{dn}^2(u;k) &= \tfrac{\mathrm{d}}{\mathrm{d}u} \mathcal{E}(u;k) \end{align*}\]
\[\begin{align*} \mathrm{sn}^2 + \mathrm{cn}^2 &= 1 \\ m\mathrm{sn}^2 + \mathrm{dn}^2 &= 1 \end{align*}\]
\[\begin{align*} \mathrm{sn}(u + K(k); k) &= \mathrm{cd}(u; k) \\ \mathrm{cn}(u + K(k); k) &= \sqrt{1-k^2} \mathrm{sd}(u; k) \\ \mathrm{dn}(u + K(k); k) &= \sqrt{1-k^2} \mathrm{nd}(u; k) \end{align*}\]
\[\begin{align*} \mathrm{sn}(u + 2K(k); k) &= -\mathrm{sn}(u; k) \\ \mathrm{cn}(u + 2K(k); k) &= -\mathrm{cn}(u; k) \\ \mathrm{dn}(u + 2K(k); k) &= \mathrm{dn}(u; k) \end{align*}\]
\[\begin{align*} \mathrm{sn}(u;0) &= \sin u & \mathrm{sn}(u;1) &= \tanh u \\ \mathrm{cn}(u;0) &= \cos u & \mathrm{cn}(u;1) &= \mathrm{sech} u \\ \mathrm{dn}(u;0) &= 1 & \mathrm{dn}(u;1) &= \mathrm{sech} u \end{align*}\]
\[\begin{align*} \mathrm{nn}' &= & 0 \quad| && \mathrm{sn}' &= & \mathrm{cn}\mathrm{dn} \quad| && \mathrm{cn}' &= - & \mathrm{sn}\mathrm{dn} \quad| && \mathrm{dn}' &= -m & \mathrm{sn}\mathrm{cn} &\\ \mathrm{ns}' &= - & \mathrm{cs}\mathrm{ds} \quad| && \mathrm{ss}' &= & 0 \quad| && \mathrm{cs}' &= - & \mathrm{ns}\mathrm{ds} \quad| && \mathrm{ds}' &= - & \mathrm{cs}\mathrm{ns} &\\ \mathrm{nc}' &= & \mathrm{dc}\mathrm{sc} \quad| && \mathrm{sc}' &= & \mathrm{dc}\mathrm{nc} \quad| && \mathrm{cc}' &= & 0 \quad| && \mathrm{dc}' &= (1-m) & \mathrm{nc}\mathrm{sc} &\\ \mathrm{nd}' &= m & \mathrm{cd}\mathrm{sd} \quad| && \mathrm{sd}' &= & \mathrm{cd}\mathrm{nd} \quad| && \mathrm{cd}' &= (m-1) & \mathrm{nd}\mathrm{sd} \quad| && \mathrm{dd}' &= & 0 \end{align*}\]
\[\begin{align*} \mathrm{sn}(u,m) &= -i \mathrm{sc}(iu, 1-m) \\ \mathrm{cn}(u,m) &= \mathrm{nc}(iu, 1-m) \\ \mathrm{dn}(u,m) &= \mathrm{dc}(iu, 1-m) \end{align*}\]
\[\begin{align*} \mathrm{sn}(u,m) &= \tfrac{1}{k} \mathrm{sn}(ku,\tfrac{1}{m}) \\ \mathrm{cn}(u,m) &= \mathrm{dn}(ku,\tfrac{1}{m}) \\ \mathrm{dn}(u,m) &= \mathrm{cn}(ku,\tfrac{1}{m}) \end{align*}\]
Weierstrass ℘
(The "℘" character is unicode 2118.)
\(\wp(z)\), depends on either lattice \(\Lambda\), lattice generators \(\omega_1,\omega_2\), invariants \(g_2\) and \(g_3\), or half-periods \(e_1,e_2,e_3\).
\[\begin{align*} \wp(z) &= e_3 + (e_1 - e_3) \mathrm{ns}^2\left(z\sqrt{e_1 - e_3}; \sqrt{\frac{e_2 - e_3}{e_1 - e_3}}\right) \\ &= e_2 + (e_1 - e_3) \mathrm{ds}^2\left(z\sqrt{e_1 - e_3}; \sqrt{\frac{e_2 - e_3}{e_1 - e_3}}\right) \\ &= e_1 + (e_1 - e_3) \mathrm{cs}^2\left(z\sqrt{e_1 - e_3}; \sqrt{\frac{e_2 - e_3}{e_1 - e_3}}\right) \end{align*}\]
TODO