Schwarz Reflection

Abstract

Let f be an analytic function on an open set U,and let V be an open set. We shall give various criteria when f can be extended to an analytic function on U ⋃ V. The process of extending f in this way is called analytic continuation. If U, V are connected, and have in common an infinite set of points which have a point of accumulation in U ⋂ V, then an analytic continuation of f to U ⋃ V is uniquely determined. Indeed, if g is analytic on V and g = f on U ⋂ V, then g is the only such function by Theorem 1.2 of Chapter III.