ABSTRACT
In Chapters 12, 13, and 14 we developed the most important techniques of analysis except for integration. Now we shall see how they are applied in the theory of analytic functions.
Analytic functions are those which are represented by their Taylor’s series. Hence they include the familiar functions of elementary calculus. Taylor’s series provide a natural way to extend the elementary functions from R to R to functions from C to C. In the extended forms their properties are far more transparent. A great deal of information about standard problems of calculus has been obtained by studying the complex extensions of various functions. For example, it has been shown that the indefinite integral
is not a combination of the usual elementary functions, no matter how complicated. While we cannot prove such a sophisticated result as this here, we can develop some of the most important general theorems and study in some detail the ex ponential, sine, and cosine functions.
