Doubly Periodic Function

A function is said to be doubly periodic if it has two periods and whose ratio is not real. A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function (Whittaker and Watson 1990, p. 429). The periods and play the same part in the theory of elliptic functions as does the single period in the case of the trigonometric functions.

Jacobi (1835) proved that if a univariate single-valued function is doubly periodic, then the ratio of periods cannot be real, as well as the impossibility for a single-valued univariate function to have more than two distinct periods (Boyer and Merzbach 1991, p. 525).

References

Apostol, T. M. "Doubly Periodic Functions." §1.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 1-2, 1997.

Boyer, C. B. and Merzbach, U. C. A History of Mathematics, 2nd ed. New York: Wiley, 1991.

Jacobi, C. G. J. J. für Math. 13, 55-56, 1835. Reprinted in Gesammelte Werke, Vol. 2, 2nd ed. Providence, RI: Amer. Math. Soc., pp. 25-26, 1969.

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.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Doubly Periodic Function

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