Complex Analysis through Examples and Exercises
The book Complex Analysis through Examples and Exercises has come out from the lectures and exercises that the author held mostly for mathematician and physists . The book is an attempt to present the rat her involved subject of complex analysis through a
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Kluwer Text in the Mathematical Sciences VOLUME21
A Graduate-Level Book Series
The titfes published in this series are listed at the end 0/ this vofume.
Complex Analysis through Examples and Exercises by
Endre Pap Institute of Mathematics, University of Novi Sad, Novi Sad, Yugoslavia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5253-7 ISBN 978-94-017-1106-7 (eBook) DOI 10.1007/978-94-017-1106-7
Printed an acid-free paper
AII Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 N o part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents Contents
v
Preface 1
IX
The Complex Numbers
1
1.1
Algebraic Properties
1
1.1.1
Preliminaries
1
1.1.2
Examples and Exercises
2
1.2 The Topology of the Complex Plane
2
......
1.2.1
Preliminaries
1.2.2
Examples and Exercises
Sequences and series
2.1
Sequences
.....
32 33 37
37
2.1.1
Preliminaries
37
2.1.2
Examples and Exercises
38
2.2 Series . . . . . . . . .
3
32
44
2.2.1
Preliminaries
44
2.2.2
Examples and Exercises
45
Complex functions
53
3.1
General Properties
53
3.1.1
Preliminaries
53
3.1.2
Examples and Exercises
54
vi
CONTENTS
3.2 Special Functions ..
64
3.2.1
Preliminaries
64
3.2.2
Examples and Exercises
65
3.3 Multi-valued functions
4
4.2
5
Preliminaries
68
3.3.2
Examples and Exercises
68 73
.......
73
4.1.1
Preliminaries
73
4.1.2
Examples and Exercises
73
Basics
Special mappings
..
74
4.2.1
Preliminaries
74
4.2.2
Examples and Exercises
75
The Integral
5.1
6
3.3.1
Conformal mappings
4.1
68
Basics
103 • • •
0
103
••
5.1.1
Preliminaries
103
5.1.2
Examples and Exercises
104
The Analytic functions
129
6.1
129
The Power Series Representation
......
6.1.1
Preliminaries
6.1.2
Examples and Exercises
6.2 Composite Examples ...... 7 Isolated Singularities
7.1
129 131 162 171
Singularities . . . .
171
7.1.1
Preliminaries
171
7.1.2
Examples and Exercises
172
7.2 Laurent series .... 7.2.1
Preliminaries
177 177
CONTENTS
vii
7.2.2
Examples and Exercises . . . . . . . . . . . . . . . . . . . . . 179
Residues
191
8
8.1
8.2 9
Residue Theorem
· 191
8.1.1
Preliminaries
· 191
8.1.2
Examples and Exercises
· 193
Composite Examples . . . . . .
Analytic continuation
9.1
9.2
Continuation
...
.206 227
.227
9.1.1
Preliminaries
.227
9.1.2
Examples and Exercises
.228
Composite Examples . . . . . .
10 Integral transforms
10.1 Analytic Functions Defined by Integrals.
.233 255
.255
10.1.1 Preliminaries . . . . . .
.255
10.1.2 Examples and Exercises
.256
10.2 Composite Examples . . . . . .
.268
11 Miscellaneous Examples
313
Bibliogra
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