Power Series

Abstract

So far, we have given only rational functions as examples of holomorphic functions. We shall study other ways of defining such functions. One of the principal ways will be by means of power series. Thus we shall see that the series

$$ 1 + z + \frac{{{{z}^{2}}}}{{2!}} + \frac{{{{z}^{3}}}}{{3!}} +\cdots$$

converges for all z to define a function which is equal to e^z. Similarly, we shall extend the values of sin z and cos z by their usual series to complex valued functions of a complex variable, and we shall see that they have similar properties to the functions of a real variable which you already know.