Preface

Summary

This book aims to give a clear explanation of classical theory of analytic functions; that is, the theory of holomorphic functions of one complex variable. In modern treatments of function theory it is customary to call a function holomorphic if its derivative exists. However, we return to the old definition, calling a function holomorphic if its derivative exists and is continuous, since we believe this is a more natural approach.

The first difficulty one encounters in writing an introduction to function theory is the topology involved in Cauchy's Theorem and Cauchy's integral formula. In the first chapter of the book we prove the latter in a topologically simple case, and from that result we deduce the basic properties of holomorphic functions. In the second chapter we prove the general version of Cauchy's Theorem and integral formula. I have tried to replace the necessary topological considerations with elementary geometric considerations. This way turned out to be longer than I expected, so that in the original Japanese three-volume edition I had to end Volume 2 before Chapter 5 was completed. My original intention was to present classical many-valued analytic functions, in particular the Riemann surface of an algebraic function, and to introduce the general concept of a Riemann surface as its generalization. Now, with the appearance of the complete Japanese edition in a single volume, the link between the theory of Riemann surfaces and function theory is restored.