Jacobi Elliptic Functions - Definition in Terms of Theta Functions

Definition in Terms of Theta Functions

Equivalently, Jacobi elliptic functions can be defined in terms of his theta functions. If we abbreviate as, and respectively as (the theta constants) then the elliptic modulus k is . If we set, we have



Since the Jacobi functions are defined in terms of the elliptic modulus k(τ), we need to invert this and find τ in terms of k. We start from, the complementary modulus. As a function of τ it is

Let us first define

\ell = {1 \over 2} {1-\sqrt{k'} \over 1+\sqrt{k'}} =
{1 \over 2} {\vartheta - \vartheta_{01} \over \vartheta + \vartheta_{01}}.

Then define the nome q as and expand as a power series in the nome q, we obtain

Reversion of series now gives

Since we may reduce to the case where the imaginary part of τ is greater than or equal to 1/2 sqrt(3), we can assume the absolute value of q is less than or equal to exp(-1/2 sqrt(3) π) ~ 0.0658; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.

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