Modular Function

A function is said to be modular (or "elliptic modular") if it satisfies:

1.  is meromorphic in the upper half-plane ,

2.  for every matrix  in the modular group Gamma,

3. The Laurent series of  has the form

(Apostol 1997, p. 34). Every rational function of Klein's absolute invariant  is a modular function, and every modular function can be expressed as a rational function of  (Apostol 1997, p. 40). Modular functions are special cases of modular forms, but not vice versa.

An important property of modular functions is that if  is modular and not identically 0, then the number of zeros of  is equal to the number of poles of  in the closure of the fundamental region  (Apostol 1997, p. 34).

REFERENCES:

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997.

Askey, R. "Ramanujan and Hypergeometric and Basic Hypergeometric Series." In Ramanujan International Symposium on Analysis: Proceedings of the Ramanujan Birth Centenary Year International Symposium held in Pune, December 26-28, 1987 (Ed. N. K. Thakare, K. C. Sharma and T. T. Raghunathan). New Delhi: Macmillan of India, pp. 1-83, 1989.

Borwein, J. M. and Borwein, P. B. "Elliptic Modular Functions." §4.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112-116, 1987.

Rademacher, H. "Zur Theorie der Modulfunktionen." J. reine angew. Math. 167, 312-336, 1932.

Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977.

Schoeneberg, B. Elliptic Modular Functions: An Introduction. Berlin: New York: Springer-Verlag, 1974.

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