## Elliptic Function

A doubly periodic function with periods and such that

 (1)

which is Analytic and has no singularities except for Poles in the finite part of the Complex Plane. The ratio must not be purely real. If this ratio is real, the function reduces to a singly periodic function if it is rational and a constant if the ratio is irrational (Jacobi, 1835). and are labeled such that . 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 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 (the order'') equals the number of Poles of .

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 Residue 0 are called Weierstraß Elliptic Functions. Elliptic functions with two simple Poles having residues and 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

 (2)

and Weierstraß Elliptic Functions arise as solutions to differential equations of the form
 (3)

See also Elliptic Curve, Elliptic Integral, Jacobi Elliptic Functions, Liouville's Elliptic Function Theorem, Modular Form, Modular Function, Neville Theta Function, Theta Function, Weierstraß Elliptic Functions

References

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Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, 1961.

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King, L. V. On the Direct Numerical Calculation of Elliptic Functions and Integrals. Cambridge, England: Cambridge University Press, 1924.

Lang, S. Elliptic Functions, 2nd ed. New York: Springer-Verlag, 1987.

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Murty, M. R. (Ed.). Theta Functions. Providence, RI: Amer. Math. Soc., 1993.

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Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. Elliptic Function Identities.'' §1.8 in A=B. Wellesley, MA: A. K. Peters, pp. 13-15, 1996.

Whittaker, E. T. and Watson, G. N. Chs. 20-22 in A Course of Modern Analysis, 4th ed. Cambridge, England: University Press, 1943.