A FAMILY OF COMPACTIFICATIONS ACCOUNTING FOR ALL ARGUMENTS OF INFINITY YOTAM I. GINGOLD AND HARRY GINGOLD Abstract. We study a family of compactifications of the complex plane that distinguishes among the values of positive infinity, negative infinity, and other “arguments” of infinity. We augment the complex plane with an ideal set of points on an ideal circle. Each point on this circle corresponds to a differ- ent ray in the complex plane emanating from the origin. In this manner we obtain the “ultra extended complex plane.” The set of points in the ultra extended complex plane maps to a bowl-shaped subset of the Riemann sphere via a certain projection. We obtain the Riemann stereographic projection as a degenerate limit of a family of projections. Thus, we demonstrate how the infinitely many different directions at infinity degenerate into a single ideal point at infinity that is added to R 2 in order to produce the extended complex plane. The features of this mapping are studied; the introduction of a metric on the ultra extended complex plane is a focal subject of this paper. 1. Introduction The stereographic projection provides a one to one mapping between the points of the extended complex plane and the points on the Riemann sphere. Basic text books in complex analysis, e.g. [2, 6], inform us that the stereographic projection is useful for many purposes and serves as an important tool for the concrete visualization of the extended complex plane. The image of the ideal point infinity, the north pole of the Riemann sphere, is treated like the image of any other point in the complex plane. The metric induced by the stereographic projection is well exploited in the theory of meromorphic and entire functions. The fundamental idea is endemic to topology and a useful tool in combinatorial topology (e.g. see [5]). However, the stereographic projection does not distinguish between positive infinity or negative infinity, or among other, different “values” of infinity. It is important both in mathematics and mathematical physics to possess a mean that will distinguish between various “arguments of infinity.” Our work addresses this issue by proposing a family of projections in which the stereographic projection becomes a particular, albeit degenerate, case. We study a family of compactifications of the complex plane that takes into account the argument of a point with infinite magnitude. We augment the complex plane C with an ideal set of points, ID = {∞(cos θ, sin θ) | 0 ≤ θ< 2π}, on an ideal circle. Each point on this circle corresponds to a different ray emanating from the origin. In this manner we obtain the “ultra extended complex plane,” Date : September 17, 2003 . 1991 Mathematics Subject Classification. Primary 30A99; Secondary 30C65. Key words and phrases. stereographic projection, compactification, argument, infinity, bijec- tion, metric, conformal . 1