Introduction

The study of singularities of analytic functions can be considered

area of the theory of functions of several complex variables and

braic/analytic geometry. It has in the meantime, together with th

of singularities of differentiable mappings, developed into an ind

subject, singularity theory. Through its connections with very ma

mathematical areas and applications to natural and economic scie

in technology (for example, under the heading 'catastrophe theo

theory has aroused great interest. The particular appeal, but also

ticular difficulty, lies in the fact that deep results and methods from

branches of mathematics come into play here.

The aim of this book is to present the foundations of the theory

tions of several complex variables and on this basis to develop the fu

tal concepts of the theory of isolated singularities of holomorphic

systematically. It is derived from lectures given by the author to

matics students in their third and fourth year to introduce them to

research questions in the area of the theory of functions of several v

The book has its genesis in this. As prerequisites we assume only

ductory knowledge of the theory of functions of a single complex

and of algebra, such as students will normally acquire in their first t

of study. The first two chapters correspond to a continuation of th

on complex analysis and deal with Riemann surfaces and the theory

tions of several complex variables. They also present an introduction

complex geometry. In the third chapter the results will be applied

mation and classification of isolated singularities of holomorphic f

These three chapters have grown from notes for the author's lec

Riemann surfaces and the theory of functions of several complex