Functions on the plane and on the sphere

Abstract

By adding a “point at infinity”, denoted ∞, to the complex plane, we obtain the Riemann sphere (III.1); it allows an elegant description of meromorphic and, in particular, rational functions and an interpretation of Möbius transformations as automorphisms of the sphere (III.2,4). Important theorems about functions that are holomorphic on all of c (“entire functions”) follow from the fact that the point ∞ is an isolated singularity of these functions (III.3). Polynomials and rational functions are investigated in detail in III.2; in particular, this section contains proofs of the fundamental theorem of algebra as well as historical notes. With the logarithm function and the functions that arise from it, we conclude our “elementary” study of the elementary functions; among other things, we describe the local mapping properties of holomorphic or meromorphic functions via root functions. Partial fraction decompositions (III.6) are an essential tool in the study of meromorphic functions; in addition to a general existence theorem, this section contains the decompositions of the functions cot πz and 1/ sin πz and their consequences. The Weierstrass product formula (III.7) for entire functions substantially generalizes the factorisation of polynomials into linear factors. We shall use it in V.1 and V.4 to define non-elementary functions.