Elliptic function

An elliptic function on the complex numbers is a function of the form

E(z; a,b) = ∑_i∑_j(z-ia-bj)^-2

where a and b are complex parameters and i and j range over the integers. As written, this series is improper and divergent; but it can be made convergent by taking the Cauchy principal value, which is the limit as x->&inf; of the sum of those terms with |z-ia-bj|<x. The function is periodic with two periods, a and b. Plotting E(z) on x versus E'(z) on y results in an elliptic curve.

A real elliptic function can also be defined in the same way. Either a is real and b imaginary (in which case the elliptic curve has two parts, E(z+b/2) being also real for real z) or a+b is real and a-b is imaginary (in which case the elliptic curve has one part).

Degenerate elliptic functions and curves are obtained by setting a or b to infinity. If a or b is infinite, but not both, the Cauchy principal value diverges and other means must be used to define the function. If both are infinite, E(z) is simply 1/z². If a is real and b is infinite, the curve consists of one smooth part and one point. If a is imaginary and b is infinite, the curve is a loop that crosses itself. If both are infinite, the curve is the semicubical parabola x³=y²/64.