Introduction

In this chapter we introduce the concept of differentiation of a complex function of a complex variable. There are several ways of approaching this topic, and we shall consider at least two. Given that a complex number may be regarded as a pair of real numbers, we need to make very clear the distinction between differentiability of a function that has two real arguments and differentiability of a function with a single complex argument, and shall take as our starting point a review of the differentiation of functions of two real variables. This approach has the merit of generalizing in a straight-forward way to functions of many real or complex variables. We shall also consider another simple approach to differentiation based on the limit of a ratio. This latter approach is perhaps more familiar if you have taken a course in one-variable real calculus, but does not generalize to functions of several real or complex variables. A key result that we will establish is that a complex function is differentiable in the complex sense if (a) it is differentiable when considered as two real functions of two real variables and also (b) the Cauchy—Riemann equations (partial differential equations) apply. These equations link the real and imaginary parts of the function. After proving some basic results about complex differentiability, e.g., the product, quotient and chain rules, we then derive one of the principal results of basic complex analysis — that power series are differentiable within their radius of convergence.