Elliptic Function - Weierstrass's Elliptic Functions

Weierstrass's Elliptic Functions

With the definition of elliptic functions given above (which is due to Weierstrass) the Weierstrass elliptic function is constructed in the most obvious way: given a lattice as above, put

This function is clearly invariant with respect to the transformation for any . The addition of the terms is necessary to make the sum converge. The technical condition to ensure that an infinite sum such as this converges to a meromorphic function is that on any compact set, after omitting the finitely many terms having poles in that set, the remaining series converges normally. On any compact disk defined by, any satisfies

and it can be shown that the sum converges regardless of .

By writing as a Laurent series and explicitly comparing terms, one may verify that it satisfies the relation

where

and .

This means that the pair parametrize an elliptic curve.

The functions take different forms depending on, and a rich theory is developed when one allows to vary. To this effect, put and, with . (After a rotation and a scaling factor, any lattice may be put in this form.)

A holomorphic function in the upper half plane which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular form. That is, a holomorphic function is a modular form if

for all .

One such function is Klein's j-invariant, defined by

where and are as above.