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Elliptic Function

An elliptic function is a doubly periodic function with periods 2omega_1 and 2omega_2 such that

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

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 Im[tau]=Im[omega_2/omega_1]>0, where Im[z] is the imaginary part.

A "cell" of an elliptic function is a fundamental period parallelogram in the complex plane, with opposite sides identified. Roots and poles are counted with their multiplicities, with each boundary point counted only once modulo the periods. Properties obeyed by nonconstant 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. If a_1,...,a_r and b_1,...,b_r are the affixes of the roots and poles in a cell, repeated according to their multiplicities, then sum_(j=1)^(r)a_j-sum_(j=1)^(r)b_j=2momega_1+2nomega_2 for some integers m and n. Thus the two sums are equal modulo the period lattice.

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)/(dt^2)=A+Bx+Cx^2+Dx^3,
(2)

with A=C=0. For example, x(t)=sn(t,k) satisfies x^('')=-(1+k^2)x+2k^2x^3 (NIST, §22.13). The Weierstrass elliptic functions arise as solutions to differential equations of the form

(d^2x)/(dt^2)=A+Bx+Cx^2,
(3)

with A=-g_2/2, B=0, and C=6. In particular, x(t)=P(t+a;g_2,g_3) satisfies x^('')=6x^2-g_2/2, with g_3 determined by the first integral (x^')^2=4x^3-g_2x-g_3 (NIST, §23.3).

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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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, Germany: Springer-Verlag, 1971.Cayley, A. An Elementary Treatise on Elliptic Functions, 2nd ed. London, England: G. Bell, 1895.Chandrasekharan, K. Elliptic Functions. Berlin, Germany: 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, England: 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.National Institute of Standards and Technology. "Derivatives and Differential Equations." §22.13 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/22.13.National Institute of Standards and Technology. "Differential Equations." §23.3 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/23.3.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, Germany: 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. https://www2.math.upenn.edu/~wilf/AeqB.html.Prasolov, 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.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.

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Elliptic Function

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Weisstein, Eric W. "Elliptic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticFunction.html

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