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Analytic Continuation

  • Steven G. Krantz

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 f1f2 on U, then f1f2 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Steven G. Krantz
    • 1
  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA

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