Analytic Continuation Along Curves

Abstract

In this chapter we give further means to extend the domain of definition of an analytic function. We shall apply Theorem 1.2 of Chapter III in the following context. Suppose we are given an analytic function f of an open connected set U. Let V be open and connected, and suppose that U ∩ V is not empty, so is open. We ask whether there exists an analytic function g on V such that f = g on U ∩ V, or only such that f(z) = g(z) for all z in some set of points of U ∩ V which is not discrete. The above-mentioned Theorem 1.2 shows that such a function g if it exists is uniquely determined. One calls such a function g a direct analytic continuation of f, and we also say that (g, V) is a direct analytic continuation of (f, U). We use the word “direct” because later we shall deal with analytic continuation along a curve and it is useful to have an adjective to distinguish the two notions. For simplicity, however, one usually omits the word “direct” if no confusion can result from this omission. If a direct analytic continuation exists as above, then there is is unique analytic function h on U ∪ V such that h = f on U and h = g on V.