Modular Function

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

1. f is meromorphic in the upper half-plane H,

2. f(Atau)=f(tau) for every matrix A in the modular group Gamma,

3. The Laurent series of f has the form


(Apostol 1997, p. 34). Every rational function of Klein's absolute invariant J is a modular function, and every modular function can be expressed as a rational function of J (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 f is modular and not identically 0, then the number of zeros of f is equal to the number of poles of f in the closure of the fundamental region R_Gamma (Apostol 1997, p. 34).

See also

Dirichlet Series, Elliptic Function, Elliptic Lambda Function, Klein's Absolute Invariant, Modular Equation, Modular Form, Modular Group Gamma, Modular Group Gamma0, Modular Group Lambda

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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."

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Modular Function

Cite this as:

Weisstein, Eric W. "Modular Function." From MathWorld--A Wolfram Web Resource.

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