See also Dirichlet Series ,
Elliptic Function ,
Elliptic Lambda Function ,
Klein's Absolute Invariant ,
Modular
Equation ,
Modular Form ,
Modular
Group Gamma0 ,
Modular Group Gamma ,
Modular Group Lambda
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References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. 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, India: 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. Rademacher, H. "Zur Theorie
der Modulfunktionen." J. reine angew. Math. 167 , 312-336, 1932. Rankin,
R. A. Modular
Forms and Functions. Schoeneberg,
B. Elliptic
Modular Functions: An Introduction. Weisstein, E. W. "Books about Modular Functions."
http://www.ericweisstein.com/encyclopedias/books/ModularFunctions.html . Referenced
on Wolfram|Alpha Modular Function
Cite this as:
Weisstein, Eric W. "Modular Function."
From MathWorld https://mathworld.wolfram.com/ModularFunction.html
Subject classifications
is meromorphic in the upper
half-plane 
,
for every matrix 
in the modular group Gamma,
has the form
is a modular function, and every modular function can be expressed
as a rational function of 
(Apostol 1997, p. 40). A modular function is a meromorphic modular form of weight 0, and may have poles.
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).
