Complex Analysis

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Springer Science & Business Media, Jun 29, 2013 - Mathematics - 370 pages
The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recom mend to anyone to look through them. More recent texts have empha sized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex anal ysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e. g. , for differential equations. I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes. The book essentially decomposes into two parts.
 

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Contents

CHAPTER
3
1 Definition
11
4 Limits and Compact Sets
17
5 Complex Differentiability
28
CHAPTER VIII
32
7 Angles Under Holomorphic Maps
34
2 Convergent Power Series
49
3 Relations Between Formal and Convergent Series
62
2 Evaluation of Definite Integrals
180
CHAPTER VII
196
3 The Upper Half Plane
203
5 Fractional Linear Transformations
215
Harmonic Functions
224
2 Examples
234
3 Basic Properties of Harmonic Functions
241
5 The Poisson Representation
249

4 Analytic Functions
69
6 The Local Maximum Modulus Principle
79
CHAPTER III
87
2 Integrals Over Paths
94
3 Local Primitive for a Holomorphic Function
103
4 Another Description of the Integral Along a Path
109
5 The Homotopy Form of Cauchys Theorem
115
CHAPTER IV
123
3 Artins Proof
137
CHAPTER V
144
2 Laurent Series
151
4 Dixons Proof of Cauchys Theorem
162
2 The Effect of Small Derivatives
260
4 The PhragmenLindelöf and Hadamard Theorems
268
CHAPTER X
276
3 Functions of Finite Order
286
CHAPTER XI
292
3 The Addition Theorem
299
CHAPTER XII
307
CHAPTER XIII
324
CHAPTER XIV
340
Appendix
359
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