Summation Methods

  • Carlos A. Berenstein
  • Roger Gay


Given a power series \( f\left( z \right) = \sum\nolimits_{n \geqslant 0} {{a_n}} {z^n}\) of radius of convergence R, 0 < R < ∞, we are trying to find explicitly the analytic continuation of f to the largest domain, star-shaped with respect to the origin, to which f admits an analytic continuation. Let us denote by D(f) that domain. (Why is it well defined?) We shall obtain D(f) as the union of certain domains B ρ (f),such that in each of them we shall be able to describe explicitly the analytic continuation of f, these domains are parametrized by ρ ≥ 1. The domain D(f) is called the star of holomorphy of f. We start by explaining how to determine B(f) = B1 (f),usually called the Borel polygon of f.


Holomorphic Function Entire Function Analytic Continuation Finite Type Exponential Type 
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Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Carlos A. Berenstein
    • 1
  • Roger Gay
    • 2
  1. 1.Mathematics Department and Institute for Systems ResearchUniversity of MarylandCollege ParkUSA
  2. 2.Centre de Recherche en MathématiquesUniversité de Bordeaux ITalence (cedex)France

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