Modular function
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In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details.
Formally, a function f is called modular or a modular function iff it satisfies the following properties:
- f is meromorphic in the open upper half-plane H.
- For every matrix M in the modular group Γ, f(Mτ) = f(τ).
- The Laurent series of f has the form
It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant j(τ), and that every rational function of j(τ) is a modular function; furthermore, all analytic modular functions are modular forms, although the converse does not hold. If a modular function f is not identically 0, then it can be shown that the number of zeroes of f is equal to the number of poles of f in the closure of the fundamental region RΓ.
[edit] Other uses
There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.
[edit] References
- Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0
- Robert A. Rankin, Modular forms and functions, (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X