Applications of Cauchy’s Integral Formula

Abstract

In this chapter, we return to the ideas of Theorem 7.3 of Chapter III, which we interrupted to discuss some topological considerations about winding numbers. We come back to analysis. We shall give various applications of the fact that the derivative of an analytic function can be expressed as an integral. This is completely different from real analysis, where the derivative of a real function often is less differentiable than the function itself. In complex analysis, one can exploit the phenomenon in various ways. For instance, in real analysis, a uniform limit of a sequence of differentiable functions may be only continuous. However, in complex analysis, we shall see that a uniform limit of analytic functions is analytic.