217–218. Functions of a complex variable

217. Functions of a complex variable.

In Ch. III we defined the complex variable \[z = x + iy,\] and we considered a few simple properties of some classes of expressions involving \(z\), such as the polynomial \(P(z)\). It is natural to describe such expressions as functions of \(z\), and in fact we did describe the quotient \(P(z)/Q(z)\), where \(P(z)\) and \(Q(z)\) are polynomials, as a ‘rational function’. We have however given no general definition of what is meant by a function of \(z\).

It might seem natural to define a function of \(z\) in the same way as that in which we defined a function of the real variable \(x\), to say that \(Z\) is a function of \(z\) if any relation subsists between \(z\) and \(Z\) in virtue of which a value or values of \(Z\) corresponds to some or all values of \(z\). But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if \(z\) is given, so are \(x\) and \(y\), and conversely: to assign a value of \(z\) is precisely the same thing as to assign a pair of values of \(x\) and \(y\). Thus a ‘function of \(z\)’, according to the definition suggested, is precisely the same thing as a complex function \[f(x, y) + ig(x, y),\] of the two real variables \(x\) and \(y\). For example \[x – iy,\quad xy,\quad |z| = \sqrt{x^{2} + y^{2}},\quad {am}\ z = \arctan(y/x)\] are ‘functions of \(z\)’. The definition, although perfectly legitimate, is futile because it does not really define a new idea at all. It is therefore more convenient to use the expression ‘function of the complex variable \(z\)’ in a more restricted sense, or in other words to pick out, from the general class of complex functions of the two real variables \(x\) and \(y\), a special class to which the expression shall be restricted. But if we were to attempt to explain how this selection is made, and what are the characteristic properties of the special class of functions selected, we should be led far beyond the limits of this book. We shall therefore not attempt to give any general definitions, but shall confine ourselves entirely to special functions defined directly.

218.

We have already defined polynomials in \(z\) (§ 39), rational functions of \(z\) (§ 46), and roots of \(z\) (§ 47). There is no difficulty in extending to the complex variable the definitions of algebraical functions, explicit and implicit, which we gave (§§ 26–27) in the case of the real variable \(x\). In all these cases we shall call the complex number \(z\), the argument (§ 44) of the point \(z\), the argument of the function \(f(z)\) under consideration. The question which will occupy us in this chapter is that of defining and determining the principal properties of the logarithmic, exponential, and trigonometrical or circular functions of \(z\). These functions are of course so far defined for real values of \(z\) only, the logarithm indeed for positive values only.

We shall begin with the logarithmic function. It is natural to attempt to define it by means of some extension of the definition \[\log x = \int_{1}^{x} \frac{dt}{t}\quad (x > 0);\] and in order to do this we shall find it necessary to consider briefly some extensions of the notion of an integral.