Power Series, An Introduction

Introduction

If f is a real valued function defined in a neighborhood about 0, it has a taylor approximation of degree n, i.e. a polynomial of degree n whose derivatives agree with f at 0. As n approaches infinity the taylor approximations form the taylor series. If the error term approaches 0 then the taylor series approaches f. Read more on the taylor series here.

The power series is a generalization of the taylor series. Here f is a complex function of a complex variable z, and f is analytic at 0. (Remember that analytic at 0 means differentiable in a region about 0; that's the definition of analytic.) We will build polynomial approximations to f using the derivatives of f at 0. The union of all these polynomial approximations is the power series for f. It's really a taylor series; we're just using complex variables.

A real valued function, even an infinitely differentiable real valued function, need not equal its power series, but an analytic function always does. This is the power of complex differentiation. It locks everything down, and makes the function infinitely differentiable, and makes it equal to its power series, at least for a region about 0.

Through these pages, we will expand f about the origin. If you would like to expand f about p, evaluate the derivatives of f at p and write the power series in terms of z-p instead of z. This is the taylor expansion of f at p. There is no new math here, just a shift of the complex plane; so we may as well expand about the origin for notational convenience.

The first task is to show that the power series, derived from f, actually converges to f.