The word "function" expresses one of the most significant ideas in the entire history of mathematics. Yet to Leibniz ( \(1647\) - \(1716\) ), who first used the word "function", and to the mathematicians of the eighteenth century (Euler in particular), the idea of function was identified with "explicit formula"; to say " \(w\) is a function of \(z\) " meant that was given by an explicit expression in \(z\) . What, precisely, constitutes an "explicit expression" no one quite knew. The necessity for a more critical examination of the function concept was not appreciated until the nineteenth century when it was discovered that a relation between two real variables defined arbitrarily by different "functions" (in the \(18^\text{th}\) century sense) in different intervals could equally well be defined by a Fourier series. Again, the process of integration was seen to lead from rather simple functions in the sense of Euler to expressions which could not be represented by an explicit formula but nonetheless had all the properties which were expected of functions such as continuity and differentiability. Such considerations directed the growth of the concept of functions into the very general and abstract formulation of Cauchy and Dirichlet.

On the other hand, as anticipated by Lagrange in the late \(18^\text{th}\) century, there emerged a much narrower idea of function closely related to Euler’s notion of "explicit expression" and of utmost importance to all of mathematics. This was the concept of analytic function . Finally, by extending this idea to complex variables the notion of analytic function was brought out in complete clarity and placed in its proper perspective.

We begin with the most general concept of functions and then specialize the idea to that of analytic functions.

Let \(D\) be any point set in the complex plane and let \(z = x + iy\) range over the points of \(D\) . A (single-valued) function of \(\boldsymbol{z}\) in \(\boldsymbol{D}\) is a correspondence or a rule which assigns a unique complex value \(w\) to each point \(z\) of \(D\) . Given such a correspondence we write \[w=f(z)\] and say that \(f(z)\) is a function defined over the point set \(D\) . We call \(z\) the argument of the function of simply the independent variable and the point set \(D\) is then called the domain of the independent variable. We shall consider only such functions as are defined over domains \(D\) in the function theoretical sense; i.e. open connected point sets.

Considered in terms of its real and imaginary parts a complex function is equivalent to an ordered pair of real functions, the coordinates \((x, y)\) of the points of the domain \(D\) of the argument. This fact is expressed by the notation \[w=u(x, y)+i v(x, y) .\] So, from any pair of coordinates \((x, y)\) of the domain \(D\) of the \(z\) -plane we derive a pair of coordinates \((u, v)\) which may be used to represent a point of the \(z\) -plane or of a duplicate \(w\) -plane. The set of points with the coordinates \((u, v)\) is called the image or map of \(D\) by means of the function \(f(z)\) and the function itself is called a mapping . Analytic functions, as we shall see, have the significant property of mapping domains onto domains.

The scope of the definition of function is so broad that hardly any meaningful statements can be made for all functions. To obtain any significant results we must sacrifice some of this generality. What we want to obtain is a differential and integral calculus for complex functions and we are concerned only with tho functions which are subject to the operations of analysis. It is a surprising and beautiful result that the single requirement of differentiability is in itself sufficient to guarantee that a function may be repeatedly differentiated or integrated any number of times. It is this property which lends to the theory an elegant completeness which is lacking in the calculus of real functions.