Elliptic Function

A doubly periodic function with periods 2omega_1 and 2omega_2 such that


which is analytic and has no singularities except for poles in the finite part of the complex plane. The half-period ratio tau=omega_2/omega_1 must not be purely real, because if it is, the function reduces to a singly periodic function if tau is rational, and a constant if tau is irrational (Jacobi 1829). omega_1 and omega_2 are labeled such that I[tau]=I[omega_2/omega_1]>0, where I[z] is the imaginary part.

A "cell" of an elliptic function is defined as a parallelogram region in the complex plane in which the function is not multi-valued. Properties obeyed by elliptic functions include

1. The number of poles in a cell is finite.

2. The number of roots in a cell is finite.

3. The sum of complex residues in any cell is 0.

4. Liouville's elliptic function theorem: An elliptic function with no poles in a cell is a constant.

5. The number of zeros of f(z)-c (the "order") equals the number of poles of f(z).

6. The simplest elliptic function has order two, since a function of order one would have a simple irreducible pole, which would need to have a nonzero residue. By property (3), this is impossible.

7. Elliptic functions with a single pole of order 2 with complex residue 0 are called Weierstrass elliptic functions. Elliptic functions with two simple poles having residues a_0 and -a_0 are called Jacobi elliptic functions.

8. Any elliptic function is expressible in terms of either Weierstrass elliptic function or Jacobi elliptic functions.

9. The sum of the affixes of roots equals the sum of the affixes of the poles.

10. An algebraic relationship exists between any two elliptic functions with the same periods.

The elliptic functions are inversions of the elliptic integrals. The two standard forms of these functions are known as Jacobi elliptic functions and Weierstrass elliptic functions. Jacobi elliptic functions arise as solutions to differential equations of the form


and Weierstrass elliptic functions arise as solutions to differential equations of the form


See also

Doubly Periodic Function, Elliptic Curve, Elliptic Integral, Half-Period Ratio, Jacobi Elliptic Functions, Jacobi Theta Functions, Liouville's Elliptic Function Theorem, Modular Form, Modular Function, Neville Theta Functions, Weierstrass Elliptic Functions

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Elliptic Function

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