Muller J. Concepts of Function Theory 2025

Textbook in PDF format

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.
Preface.
Game Rules: Sets, Mappings, Numbers.
Basis: Limits, Elementary Functions and Metric Spaces.
Up and Down: Differentiation and Integration.
Magic Circles: Circular Functions and Local Complex Analysis.
Wonder Worlds: Global Complex Analysis.
Normal or Not: Conformal Mappings and Complex Dynamics.
Extras: Runge’s Theory and Applications