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
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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