Talk:Elliptic function

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

E(z; a,b) = ∑_m∑_n (z - m'a -n'b)^-2

where a and b are complex parameters and m and n 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->∞ of the sum of those terms with |z - m'a - n'b| < 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.

The formula for E is wrong I believe, and there are certainly other elliptic functions. I don't know how to rescue this. AxelBoldt 01:48 Nov 8, 2002 (UTC)

I just picked up the yellow book. The correct formula is

E(z; a,b) = z^-2 + ∑_m∑_n (z - m'a -n'b)^-2-(n'b)^-2,

where n=m=0 is excluded from the sum. I think it should be put at Weierstrass's elliptic function[?]. -phma