Questions tagged [modular-function]

A function is said to be Modular or, elliptic modular if it satisfies the following conditions:

$1.~~ f$ is meromorphic in the upper half-plane $H$,

$2.~~ f(\bf A\tau)=f(\tau)$ for every matrix $\bf A$ in the modular group Gamma,

$3.~~$ The Laurent series of $f$ has the form

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

A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that $f (z)$ be holomorphic in the upper half-plane. Instead, modular functions are meromorphic.

References:

https://en.wikipedia.org/wiki/Modular_form#Modular_functions

http://mathworld.wolfram.com/ModularFunction.html