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 half-period ratio  must not be purely real, because if it is, the function reduces to a singly periodic function if  is rational, and a constant if  is irrational (Jacobi 1829).  and  are labeled such that , where  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  (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 complex 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)

REFERENCES:

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

Apostol, T. M. "Elliptic Functions." §1.4 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 4-6, 1997.

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. Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, 1829.

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

Knopp, K. "Doubly-Periodic Functions; in Particular, Elliptic Functions." §9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 73-92, 1996.

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.

Oberhettinger, F. and Magnus, W. Anwendung der Elliptischen Funktionen in Physik und Technik. Berlin: Springer-Verlag, 1949.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. "Elliptic Function Identities." §1.8 in A=B. Wellesley, MA: A K Peters, pp. 13-15, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

PHelveticaov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals. Providence, RI: Amer. Math. Soc., 1997.

Siegel, C. L. Topics in Complex Function Theory, Vol. 1: Elliptic Functions and Uniformization Theory. New York: Wiley, 1988.

Venkatachaliengar, K. Development of Elliptic Functions According to Ramanujan. Technical Report, 2. Madurai Kamaraj University, Department of Mathematics, Madurai, India, n.d.

Walker, P. L. Elliptic Functions: A Constructive Approach. New York: Wiley, 1996.

Weisstein, E. W. "Books about Elliptic Functions." http://www.ericweisstein.com/encyclopedias/books/EllipticFunctions.html.

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