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A First Course on Complex Functions

  • G. J. O. Jameson

Part of the Chapman & Hall Mathematics Series book series (CHMS)

Table of contents

  1. Front Matter
    Pages i-xii
  2. G. J. O. Jameson
    Pages 1-7
  3. G. J. O. Jameson
    Pages 8-61
  4. G. J. O. Jameson
    Pages 62-113
  5. G. J. O. Jameson
    Pages 114-142
  6. Back Matter
    Pages 143-148

About this book

Introduction

This book contains a rigorous coverage of those topics (and only those topics) that, in the author's judgement, are suitable for inclusion in a first course on Complex Functions. Roughly speaking, these can be summarized as being the things that can be done with Cauchy's integral formula and the residue theorem. On the theoretical side, this includes the basic core of the theory of differentiable complex functions, a theory which is unsurpassed in Mathematics for its cohesion, elegance and wealth of surprises. On the practical side, it includes the computational applications of the residue theorem. Some prominence is given to the latter, because for the more sceptical student they provide the justification for inventing the complex numbers. Analytic continuation and Riemann surfaces form an essentially different chapter of Complex Analysis. A proper treatment is far too sophisticated for a first course, and they are therefore excluded. The aim has been to produce the simplest possible rigorous treatment of the topics discussed. For the programme outlined above, it is quite sufficient to prove Cauchy'S integral theorem for paths in star-shaped open sets, so this is done. No form of the Jordan curve theorem is used anywhere in the book.

Keywords

Cauchy's integral formula Cauchy's integral theorem Complex analysis Residue theorem Riemann surface complex number

Authors and affiliations

  • G. J. O. Jameson
    • 1
  1. 1.University of WarwickUK

Bibliographic information