A First Course on Complex Functions
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 do
- PDF / 10,910,762 Bytes
- 159 Pages / 396.85 x 612.28 pts Page_size
- 13 Downloads / 267 Views
CHAPMAN & HALL MATHEMATICS SERIES
Edited by Ronald Brown University College of North Wales Bangor J. de Wet Balliol College, Oxford
A First Course on Complex Functions G. J. O. JAMESON Lecturer in Mathematics University of Warwick
CHAPMAN AND HALL LTD 11 NEW FETTER LANE LONDON EC4
First published 1970
© 1970 G. J. O. Jameson
Softcover reprint of the hardcover 1st edition 1970
London and Colchester
ISBN-13: 978-0-412-09710-2 DOl: 10.1 007/978-94-009-5680-3
e-ISBN-13: 978-94-009-5680-3
Distributed in the U.S.A. by Barnes & Noble, Inc.
Contents
page vii
Preface Terminology and notation
Xl
Metric spaces 1. Basic theory 1.1. The complex number field 1.2. Sequences and series
1.3. 1.4. 1.5. 1.6. 1. 7.
Line segments and convexity Complex functions of a real variable Differentiation The exponential and trigonometric functions Integration
8 15 20 23
29
38 51
2. The theory of differentiable functions 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.
Cauchy's integral theorem and formula The Taylor series and its applications Entire functions and polynomials The modulus of a differentiable function Singularities; Laurent series The residue theorem Integration off' If and the local mapping theorem v
62 70 79 85 90 100 106
vi
CONTENTS
3. Further topics 3.1. 3.2. 3.3. 3.4.
The evaluation of real integrals The summation of series Partial fractions Winding numbers
page 114 128 132
137
Glossary of symbols
143
Bibliography
145
Index
147
Preface
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. The results of Complex Analysis are constantly motivated and illustrated by comparison with the real case. This policy is exemplified by the proof of Cauchy's integral theorem. Since similar formulae hold in the two cases for the integral of a derivative. it is vii
viii
PREFACE
enough to show that a differentiable function is itse
Data Loading...
