ABSTRACT
Until now in our development of transform theory, we have seen only one special kind of transform, namely transforms which correspond to discrete (possibly infinite) sums over discrete basis functions, usually the eigenfunctions of some linear operator. The extension of the ideas of transform theory to other operators requires familiarity (and some dexterity) with functions defined on the complex plane. In this chapter we start from scratch, by discussing complex numbers, complex valued functions, and complex calculus, i.e., differentiation and integration, of functions in the complex plane, as well as some of the important uses of complex variable theory to fluid motion and special functions.
