The Taylor Series: an Introduction to the Theory of Functions of a Complex Variable

Abstract

A POWER series of the form?an(z - z0)n which converges at more than one point, converges inside a circle centre z0 and coincides with the series obtained by applying Taylor's theorem to the sum-function. If another point z1 is taken inside the circle of convergence, the function can be developed in a series?bn(z - z1,)n which also converges inside a circle centre z1 the area of which may extend beyond that of the original circle of convergence. n analytic function is defined by the original series and all possible transformed series obtained in this way. Since the coefficients bn are obtained uniquely from the coefficients an it follows that the whole behaviour of the function must be determinate when the sequence of coefficients an is known. The problem of Taylor's series is therefore to deduce from a knowledge of the coefficients the behaviour of the function.