Complex AnalysisThe 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. |
Contents
| 3 | |
3 Complex Valued Functions | 12 |
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 |
4 Analytic Functions | 69 |
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 |
6 The Local Maximum Modulus Principle | 79 |
Cauchys Theorem First Part | 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 |
Other editions - View all
Common terms and phrases
a₁ absolutely convergent analytic continuation analytic function analytic isomorphism assume automorphism b₁ boundary bounded calculus Cauchy's theorem Chapter closed disc closed path coefficients complex numbers constant term contained continuous function converges absolutely converges uniformly curve defined derivative differentiable disc of radius entire function equal equations Example EXERCISES exists f be analytic f be holomorphic Figure finite number formal power series function f Hence holomorphic function homologous interval inverse isomorphism Lemma Let f Let f(z lim sup maximum modulus principle meromorphic non-zero open disc open set point of accumulation pole polynomial positive integer power series expansion primitive proves the theorem quotient radius of convergence real axis real numbers rectangle residue right-hand side say that ƒ sequence Show Suppose Theorem 1.2 unit circle unit disc upper half plane winding number write z₁