Jacobi Elliptic Functions - Expansion in Terms of The Nome

Expansion in Terms of The Nome

Let the nome be and let the argument be . Then the functions have expansions as Lambert series

\operatorname{sn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1-q^{2n+1}} \sin (2n+1)v,
\operatorname{cn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1+q^{2n+1}} \cos (2n+1)v,
\operatorname{dn}(u)=\frac{\pi}{2K} + \frac{2\pi}{K} \sum_{n=1}^\infty \frac{q^{n}}{1+q^{2n}} \cos 2nv.

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