Definition:Automorphic Function

Definition
Let $f$ be a complex function

Let $S$ be a group of transformations.

Then $f$ is automorphic $S$ :


 * $(1): \quad f$ is analytic except for poles in a region of $\C$


 * $(2): \quad$ For every $T \in S$, if $z \in D$ then $\map T z \in D$ and:
 * $\map f {\map T z} = \map T {\map f z}$