Jacobi's elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e.g. the pendulum equation). They also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. They are not the simplest way to develop a general theory, as now seen: that can be said for the Weierstrass elliptic functions. They are not, however, outmoded. They were introduced by Carl Gustav Jakob Jacobi, around 1830.

Theta functions

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

Since the Jacobi functions are defined in terms of 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

If now we set q = exp(πiτ) and expand l as a power series in 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  , we can assume the absolute value of q is less than or equal to  ; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.

Doubly-periodic functions

The three Jacobi elliptic functions are doubly periodic, meromorphic functions of z, whose periods are expressible in terms of τ and θ. If we set K = π θ² then the periods of sn are 2K and τ K, of cn are 2K and 2K + 2τ K, and of dn are K and 2 τ K. If we call the periods of cn the lattice Λ, then both sn and dn are periodic with respect to Λ, but their full lattices of periods are larger (in each case, Λ is a subgroup of index 2).

The functions satisfy the two algebraic relations

From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions