### Analytic Continuation of Functions and Uniformization of Riemann Surfaces

#### Abstract

Analytic functions on Riemann surfaces may be represented through an automorphic function on a covering. The

convergence depends on the finiteness of the Poincare series of the uniformizing group. It is known also that the harmonic measure of the ideal boundary is related to the convergence of the uniformizing Fuchsian group. This condition of a finite series representation of the function on effectively closed infinite-genus surfaces with an ideal boundary of zero harmonic measure requires a summation over

elements of the Schottky group. Since there is range in the parameter space such that the series converge, a solution to the problem of analytic continuation will allow the function to be defined over a larger domain in moduli space.

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