Summation Methods

Abstract

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.