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.