Analytic Continuation

Steven G. Krantz
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Steven G. Krantz: Washington University in St. Louis, Department of Mathematics

Chapter Chapter 10 in Handbook of Complex Variables, 1999, pp 123-141 from Springer

Abstract: Abstract Suppose that V is a connected, open subset of $$ \mathbb{C} $$ and that f1: V → $$ \mathbb{C} $$ and f2: V → $$ \mathbb{C} $$ are holomorphic functions. If there is an open, non-empty subset U of V such that f1 ≡ f2 on U, then f1 ≡ f2 on all of V (see §§3.2.3). Put another way, if we are given an f holomorphic on U, then there is at most one way to extend f to V so that the extended function is holomorphic. [Of course there might not even be one such extension: if V is the unit disc and U the disc D(3/4, 1/4), then the function f(z) = 1/z does not extend. Or if U is the plane with the non-positive real axis removed, then again no extension from U to V is possible.

Keywords: Riemann Surface; Holomorphic Function; Entire Function; Analytic Continuation; Function Element (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-1-4612-1588-2_10

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