Problems and Solutions for Complex Analysis

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Springer Science & Business Media, Oct 14, 1999 - Mathematics - 246 pages
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This book contains all the exercises and solutions of Serge Lang's Complex Analy sis. Chapters I through VITI of Lang's book contain the material of an introductory course at the undergraduate level and the reader will find exercises in all of the fol lowing topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings and har monic functions. Chapters IX through XVI, which are suitable for a more advanced course at the graduate level, offer exercises in the following subjects: Schwarz re flection, analytic continuation, Jensen's formula, the Phragmen-LindelOf theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and the Zeta function. This solutions manual offers a large number of worked out exercises of varying difficulty. I thank Serge Lang for teaching me complex analysis with so much enthusiasm and passion, and for giving me the opportunity to work on this answer book. Without his patience and help, this project would be far from complete. I thank my brother Karim for always being an infinite source of inspiration and wisdom. Finally, I want to thank Mark McKee for his help on some problems and Jennifer Baltzell for the many years of support, friendship and complicity. Rami Shakarchi Princeton, New Jersey 1999 Contents Preface vii I Complex Numbers and Functions 1 1. 1 Definition . . . . . . . . . . 1 1. 2 Polar Form . . . . . . . . . 3 1. 3 Complex Valued Functions . 8 1. 4 Limits and Compact Sets . . 9 1. 6 The Cauchy-Riemann Equations .
 

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Contents

II
1
III
3
IV
8
V
9
VI
12
VII
14
VIII
19
IX
24
XXVII
137
XXVIII
146
XXIX
153
XXX
159
XXXI
165
XXXII
167
XXXIII
175
XXXIV
179

X
28
XI
29
XII
31
XIII
36
XIV
37
XV
42
XVI
43
XVII
45
XVIII
48
XIX
51
XX
60
XXI
66
XXII
76
XXIII
93
XXIV
119
XXV
122
XXVI
126
XXXV
181
XXXVI
185
XXXVII
187
XXXVIII
191
XXXIX
198
XL
201
XLI
206
XLII
211
XLIII
213
XLIV
214
XLV
219
XLVI
223
XLVII
235
XLVIII
238
XLIX
241
L
245
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