Modular function

In mathematics, a function j is called modular or a modular function iff it satisfies the following properties:

1. j is meromorphic in the upper half plane H
2. For every matrix M in the modular group Γ, j(Mτ) = j(τ)
3. The Laurent series of j has the form

$j(\tau )=\sum _{n=-m}^{m}a(n)e^{2i\pi n\tau }$

It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant J, and that every rational function of J is a modular function; furthermore, all modular functions are modular forms, although the converse does not hold.

If a modular function j is not identically 0, then it can be shown that the number of zeroes of j is equal to the number of poles of j in the closure of the fundamental region R_Γ.

There are a number of other usages of the term modular function, apart from this classical one; for example in the theory of Haar measure there is a function Δ(g) determined by the conjugation action.