This textbook presents a direct path from real analysis of one variable to function theory. Classical topics of one-dimensional real analysis, such as differential and integral calculus, are largely presented from a complex perspective. The goal is a self-contained exposition extending to the Runge theorems and the dynamics of entire functions. Short sections appended to each chapter on concepts of function theory provide glimpses into higher-dimensional analysis and an impression of its universal significance for mathematics. The book is structured so that parts can also serve as a basis for a seminar.
Thus, this fascinating area of mathematics becomes accessible to students whose programs do not focus on mathematics and for whom a classical introduction to function theory would be too time-consuming. This book enables them to take a step into complex analysis, through which they can recognize a multitude of connections that remain hidden in real analysis.
The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.
Rules of the Game: Sets, mappings, numbers.- Basics: Limits, elementary functions, and metric spaces.- Up and Down: Differentiation and integration; Magic Circles: Circular functions and local function theory.- Wonder Worlds: Global function theory.- Normal or Not: Conformal mappings and complex dynamics.- Extras: Runge theory and applications.
Prof. Dr. Jürgen Müller is a lecturer in the Department of Mathematics at the University of Trier. He conducts research in the field of complex analysis of one variable, focusing particularly on approximation in the complex domain. His motivation for writing this textbook stemmed from the idea of demonstrating a direct approach to function theory from the fundamentals of analysis, thereby enabling an immediate entry into the theory.
