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. $f({\hbox{\sf A}}\tau)=f(\tau)$ for every Matrix A in the modular group $\Gamma$,

3. The Laurent Series of $f$ has the form

$$f(\tau)=\sum_{n=-m}^m a(n)e^{2\pi in\tau}$$

(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).

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 Elliptic Function, Modular Form

References

Modular Functions

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

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.