Automorphic Function

An automorphic function  of a complex variable  is one which is analytic (except for poles) in a domain  and which is invariant under a countably infinite group of linear fractional transformations (also known as Möbius transformations)

Automorphic functions are generalizations of trigonometric functions and elliptic functions.

REFERENCES:

Ford, L. Automorphic Functions. New York: McGraw-Hill, 1929.

Hadamard, J.; Gray, J. J.; and Shenitzer, A. Non-Euclidean Geometry in the Theory of Automorphic Forms. Providence, RI: Amer. Math. Soc., 1999.

Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton University Press, 1971.

Siegel, C. L. Topics in Complex Function Theory, Vol. 2: Automorphic Functions and Abelian Integrals. New York: Wiley, 1988.