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)
See alsoAutomorphic Form, Modular Function, Möbius Transformations, Zeta Fuchsian
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ReferencesFord, 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.
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Weisstein, Eric W. "Automorphic Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AutomorphicFunction.html