MODULAR FUNCTION

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''(''M''τ) = ''f''(τ).
# The Laurent series of ''f'' has the form
::$f(\; au)\; =\; sum\_\{n=-m\}^infty\; a(n)\; e^\{2ipi\; n\; au\}.$
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 ''R''_Γ.

Contents

Other uses

References

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.

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