Automorphic Function

An automorphic function $f(z)$ of a Complex variable $z$ is one which is analytic (except for Poles) in a domain $D$ and which is invariant under a Denumerably Infinite group of Linear Fractional Transformations (also known as Möbius Transformations)

$$z' = {az+b \over cz+d}.$$

Automorphic functions are generalizations of Trigonometric Functions and Elliptic Functions.

See also Modular Function, Möbius Transformations, Zeta Fuchsian