The Geometry of Chains on a Complex Line

Abstract

Introduction. This paper forms an elementary chapter in the projective geometry on a complex line, i. e., of a line whose points are isomorphic with the system of ordinary complex numbers and infinity. It furnishes a synthetic treatment of certain well-known topics in the theory of functions of a complex variable; it has close contact with recent work on the foundations of projective geometry; finally, it forms the basis for certain generalizations, which offer a powerful synthetic approach to various analytic problems. To this last aspect of the paper I shall return in detail on a future occasion. t The first two aspects I discuss briefly in this introduction. But first, in view of the fact that the notion of a chain of points on a line does not appear to be generally familiar, it seems desirable to describe it briefly. This is most readily done in analytic language. Any point of a projective line may be conveniently represented by a single coordinate x. If the points of a line are hereby brought into one-to-one correspondence with the ordinary real numbers and infinity, we are wont to speak of a real line; if on the other hand the points of a line are hereby made isomorphic with the system of ordinary complex numbers and infinity we speak of a complex line. A real representation of the points of a complex line may then be obtained by the usual method for representing complex numbers by the real points of an Argand plane. The points of the line with real coordinate x form a subset (E0 of points on the line in the representation mentioned this subset q0 is represented by the points of the real axis. By means of a projective transformation on the line the set Go is transformedinto another or the same set (E. Any such set of points ( on the line which is projective with the set G3 of real points on the line is called a chain on the line. IIn the Argand representation the chains on the line evidently correspond to the circles (and straight lines) of the plane.