ABSTRACT
The shortest path between two truths in the real domain passes through the complex domain.
Jacques Hadamard
We have already seen how complex functions can have significantly different properties than real-valued or vector-valued functions. Not the least of these are the very different and far-reaching implications associated with complex differentiability. The theory of complex integration is no less rich. Some of the results we ultimately develop will rely on subtle topological properties of certain subsets of the plane, as well as the curves along which we will integrate. In fact a proper development of the main theorem of this chapter, Cauchy’s integral theorem, will require quite a bit of careful effort due to such subtleties. Despite this, we will find that integration is as useful an analytical tool as differentiation for investigating complex functions and their key properties. In fact it is in the theory of complex integration where we will discover some of the most interesting and useful results in all of analysis, results that will apply even to real functions of a real variable.
