Jacobi elliptic functions

Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate $\vartheta (0;\tau )$ as $\vartheta$, and $\vartheta _{01}(0;\tau ),\vartheta _{10}(0;\tau ),\vartheta _{11}(0;\tau )$ respectively as $\vartheta _{01},\vartheta _{10},\vartheta _{11}$ (the theta constants) then the modulus k is $({\vartheta _{10} \over \vartheta })^{2}$. If we set $u=\pi \vartheta ^{2}z$, we have

$sn(u;k)=-{\vartheta \vartheta _{11}(z;\tau ) \over \vartheta _{10}\vartheta _{01}(z;\tau )}$

$cn(u;k)={\vartheta _{01}\vartheta _{10}(z;\tau ) \over \vartheta _{10}\vartheta _{01}(z;\tau )}$

$dn(u;k)={\vartheta _{01}\vartheta (z;\tau ) \over \vartheta \vartheta _{01}(z;\tau )}$

Since the Jacobi functions are defined in terms of $k(\tau )$, we need to invert this and find τ in terms of k. We start from $k'={\sqrt {1-k^{2}}}$, the complementary modulus. As a function of τ it is

$k'(\tau )=({\vartheta _{01} \over \vartheta })^{2}$

Let us first define

$l={1 \over 2}{1-{\sqrt {(}}k') \over 1+{\sqrt {(}}k')}={1 \over 2}{\vartheta -\vartheta _{01} \over \vartheta +\vartheta _{01}}$

If now we set $q=\exp(\pi iz)$ and expand l as a power series in q, we obtain

$l={1 \over 2}{q+q^{9}+q^{25}\cdots \over 1+2q^{4}+2q^{16}\cdots }$

Reversion of series now gives

$q=l+2l^{5}+15l^{9}+150l^{13}+\cdots$

which converges very rapidly and easily allows us to find the appropriate value for q.

The three Jacobi elliptic functions are doubly periodic, meromorphic functions of z, whose periods are expressible in terms of τ and the theta constants. If we set $K=\pi \vartheta ^{2}$ then the periods of sn are $2K$ and $\tau K$, of cn are $2K$ and $2K+2\tau K$, and of dn are $K$ and $2\tau K$.