Analytic Continuation and Riemann Surfaces

Abstract

Consider the following problem. Let f be an analytic function on a region G; when can f be extended to an analytic function f₁ on an open set G₁ which properly contains G? If G₁, is obtained by adjoining to G a disjoint open set so that G becomes a component of G₁,f can be extended to G₁ by defining it in any way we wish on G₁—G so long as the result is analytic. So to eliminate such trivial cases it is required that G₁ also be a region.