Definition:Automorphic Function

Definition

Let $f$ be a complex function

Let $S$ be a group of transformations.

Then $f$ is automorphic with respect to $S$ if and only if:

$(1): \quad f$ 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:

$\map f {\map T z} = \map T {\map f z}$

Also see

• Results about automorphic functions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): automorphic function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): automorphic function