The Complex Number System

Abstract

We denote the set of all real numbers by ℝ. It is assumed that each reader is acquainted with the real number system and all its properties. In particular we assume a knowledge of the ordering of ℝ, the definitions and properties of the supremum and infimum (sup and inf), and the completeness of ℝ (every non-empty set in ℝ which is bounded above has a supremum). It is also assumed that every reader is familiar with sequential convergence in ℝ and with infinite series. Finally, no one should undertake a study of Complex Variables unless he has a thorough grounding in functions of one real variable. Although it has been traditional to study functions of several real variables before studying analytic function theory, this is not an essential prerequisite for this book. There will not be any occasion when the deep results of this area are needed.