Analytic Continuation

  • Steven G. Krantz


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.


Riemann Surface Holomorphic Function Entire Function Analytic Continuation Function Element 
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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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