Conformal mapping. It is not our object in this chapter to give a general survey of the geometrical and physical applications of the theory of functions of a complex variable, but only to consider certain especially interesting cases. On the geometrical side we shall confine ourselves to the study of equations w = f(z) regarded as transformations which bring about a correspondence between points of the w- and z-planes.

In a map of a piece of the earth's surface small enough to be thought of as plane, a straight line on the earth will in general be represented by a curved line on the map, and points an inch apart in one part of the map may represent points a mile apart on the earth, while in another part of the map the scale may be two miles to the inch. We are familiar with such conditions in Mercator's charts. But it is a desideratum that each point on the map correspond to one point of the region mapped, and vice versa. That is, if we set up a uv-co-ordinate system in the map, and an xy-system in the region mapped, then u and v should be defined and single-valued throughout the xy-region, and their corresponding values should cover once, without duplication, the uv-region. If we use complex variables w = u+iv and z = x+iy, and the region S of the z-plane is mapped on the region ∑ of the w-plane, then w is a function of z since a value of w corresponds to each value of z.

Elementary operations with complex numbers. When real numbers are combined by addition, subtraction, multiplication, or division with non-vanishing divisor, the results are real numbers; such numbers therefore form a closed system for these operations. But this is not always true when we pass to root extraction. No real number can be the square root of a negative real number.

The situation is analogous to one which exists for the number system composed of the positive integers. Here we have a system closed for addition, but in which subtraction of a number from one not greater than itself is impossible. When it seems desirable to allow such an operation, the difficulty is met by the enlargement of the number system so as to include zero and the negative integers. In order that division with non-vanishing divisor may always produce a number within the system, we pass from the system of positive and negative integers to the system of rational numbers which includes fractions as well as integers. The totality of all real numbers constitutes a still more inclusive system which is closed for the additional operation of passing to a finite limit.

We shall find that complex numbers include the real numbers and form a system which permits root extraction as well as the other operations we have noted.

The first of the Carus Monographs, Professor Bliss's Calculus of Variations, has for its successor the present volume on Analytic Functions of a Complex Variable. The reader is assumed to have the same preparation as for the preceding monograph, that is to say, an acquaintance with elementary differential and integral calculus. Without such knowledge one may, however, obtain some idea of the scope and purposes of the theory of functions from the following pages. Those should profit most who are familiar with more than the elements of the calculus.

The theory of functions of a complex variable has been developed by the efforts of thousands of workers through the last hundred years. To give even the briefest account of the present state of that theory in all its branches would be impossible within the limits of this book. What is attempted here is a presentation of fundamental principles with sufficient details of proof and discussion to avoid the style of a mere summary or synopsis. In various places there are indications of directions in which special portions of the subject branch off from the main stem. For almost every topic the reader is given several references; and, be it for better or worse, there are no footnotes.

The system of references perhaps needs some defense, certainly it requires explanation. Numerous citations of authorities are favorite means for an author to show his erudition—or someone's else.

Isolated singularities. A point at which a single-valued function has not a derivative, or in every neighborhood of which there are points at which the function has not a derivative, is called a singular point of the function. An especially interesting class of such points is composed of those possessing a neighborhood throughout which the function is analytic but which, of course, does not include the point itself. A point answering to this description is an isolated singular point.

It is useful here, for purposes of comparison, to consider the kinds of isolated discontinuities which single-valued real functions of a real variable x may possess. Such a function may, first of all, be not defined at a point x = a; or its value at x = a may be given so that, although f(x) has a finite limit A as x approaches a, we do not have A = f(a). Thus if f(x) = 0 except at x = 0, then f(x) will be discontinuous for x = 0 unless we add to our definition that f(x) = 0 at x = 0. If f(x) has a finite limit A as x approaches a, a discontinuity due to no definition or an inappropriate definition of f(a) is called a removable or artificial discontinuity. It remains to consider discontinuities due to the fact that f(x) has no finite limit when x approaches a.