Elliptic Function

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

$$f(z+2\omega_1)=f(z+2\omega_2)=f(z),$$ (1)

which is Analytic and has no singularities except for Poles in the finite part of the Complex Plane. The ratio $\omega_1/\omega_2$ 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). $\omega_1$ and $\omega_2$ are labeled such that $\Im(\omega_2/\omega_1)>0$. 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 $f(z)-c$ (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 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

$${d^2x\over dt^2}=A+Bx+Cx^2+Dx^3,$$ (2)

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

$${d^2x\over dt^2}=A+Bx+Cx^2.$$ (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

Elliptic Functions

Akhiezer, N. I. Elements of the Theory of Elliptic Functions. Providence, RI: Amer. Math. Soc., 1990.

Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

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

Byrd, P. F. and Friedman, M. D. Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev. Berlin: Springer-Verlag, 1971.

Cayley, A. An Elementary Treatise on Elliptic Functions, 2nd ed. London: G. Bell, 1895.

Chandrasekharan, K. Elliptic Functions. Berlin: Springer-Verlag, 1985.

Du Val, P. Elliptic Functions and Elliptic Curves. Cambridge, England: Cambridge University Press, 1973.

Dutta, M. and Debnath, L. Elements of the Theory of Elliptic and Associated Functions with Applications. Calcutta, India: World Press, 1965.

Eagle, A. The Elliptic Functions as They Should Be: An Account, with Applications, of the Functions in a New Canonical Form. Cambridge, England: Galloway and Porter, 1958.

Greenhill, A. G. The Applications of Elliptic Functions. London: Macmillan, 1892.

Hancock, H. Lectures on the Theory of Elliptic Functions. New York: Wiley, 1910.

Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntraeger, 1829.

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.

Lawden, D. F. Elliptic Functions and Applications. New York: Springer Verlag, 1989.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 427 and 433-434, 1953.

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

Neville, E. H. Jacobian Elliptic Functions, 2nd ed. Oxford, England: Clarendon Press, 1951.

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.