Analytic Continuation

Abstract

The last two sections contained proofs of the local existence of analytic functions satisfying either an analytic equation or an analytic linear differential equation. For each point a in a connected open set G they provided a number of analytic functions ga satisfying the equation on some disk with center a. The problem of analytic continuation is to decide whether, for each a, it is possible to choose one of the ga so that they fit together to determine an analytic function on all of G. If, for example, G is the complement of 0, and the La are the branches of the logarithm, it is not possible to piece them together to form an analytic function on G: if it were, 1/z would have a primitive on G, so its integral over a circle around 0 would have to be 0. More generally, the problem of analytic continuation is as follows: given a connected open set G, a point a ∈ G, and a function f, analytic on a neighborhood of a, is there an F, analytic on G and coinciding with f on a neighborhood of a? It is approached via the study of analytic continuation along paths.