Complex Analysis in One VariableThe original edition of this book has been out of print for some years. The appear ance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhauser. Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal. Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes. |
Contents
| 3 | |
| 6 | |
4 | 41 |
Covering Spaces and the Monodromy Theorem | 52 |
5 | 63 |
3 | 65 |
The Winding Number and the Residue Theorem | 69 |
Inhomogeneous CauchyRiemann Equation and Runges Theorem | 97 |
1 | 239 |
8 | 240 |
Baires Theorem | 253 |
Elementary Theory of Holomorphic Functions | 267 |
43 | 268 |
60 | 274 |
Radius of convergence of power series | 275 |
Fundamental properties of holomorphic functions | 282 |
Picards Theorem | 129 |
Riemann Mapping Theorem and Simple Connectedness in the Plane | 138 |
4 | 146 |
5 | 151 |
Compact Riemann Surfaces | 161 |
4 | 175 |
6 | 179 |
10 | 184 |
The Corona Theorem | 186 |
Subharmonic Functions and the Dirichlet Problem | 209 |
36 | 234 |
Complex square roots in celestial mechanics | 288 |
Covering Spaces and the Monodromy Theorem | 296 |
The Inhomogeneous CauchyRiemann Equation and Runges Theorem | 315 |
267 | 325 |
Applications of Runges Theorem | 331 |
297 | 338 |
Functions of Several Complex Variables | 343 |
Compact Riemann Surfaces | 351 |
6 | 358 |
Subharmonic Functions and the Dirichlet Problem | 368 |
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Common terms and phrases
analytic continuation analytic isomorphism bounded Cauchy's theorem choose closed curve closed rectangle compact set compact subset complex numbers complex variables connected component connected open set constant containing continuous function converges uniformly convex covering map Definition denote disc dx dy endpoints equation Exercise exists f dz finite fractional linear transformation function f ƒ dz germ Hı(U Hence holomorphic functions holomorphic map homeomorphism homotopy Im(y integral Lemma Let f let ƒ manifolds Math maximum principle meromorphic function neighborhood nonempty open mapping theorem orda f Picard theorem piecewise differentiable curve poles polynomial primitive of ƒ Proposition prove r₁ Reit relatively compact Remark residue theorem Riemann surface Runge's theorem sequence simply connected subharmonic functions Suppose theory Weierstrass y₁ Zı(U zeros Σπί ди ду