Functions of One Complex Variable I

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Springer Science & Business Media, Aug 24, 1978 - Mathematics - 317 pages
4 Reviews
This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.
  

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It is a great book. Clear writing and good exercise collection but a bit hard to follow.

Contents

The Complex Number System
1
6 The extended plane and its spherical representation
8
Elementary Properties and Examples of Analytic Functions
30
Complex Integration
58
Entire Functions
77
Singularities
103
The Maximum Modulus Theorem
128
Compactness and Convergence in
142
Analytic Continuation and Riemann Surfaces
210
Harmonic Functions
252
Harmonic Functions Redux
253
Potential Theory in the Plane
276
The Range of an Analytic Function
292
Calculus for Complex Valued Functions
303
References
311
List of Symbols
317

Runges Theorem
195

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