Complex Variables: An IntroductionTextbooks, even excellent ones, are a reflection of their times. Form and content of books depend on what the students know already, what they are expected to learn, how the subject matter is regarded in relation to other divisions of mathematics, and even how fashionable the subject matter is. It is thus not surprising that we no longer use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our courses. The last two decades have seen a significant change in the techniques used in the theory of functions of one complex variable. The important role played by the inhomogeneous Cauchy-Riemann equation in the current research has led to the reunification, at least in their spirit, of complex analysis in one and in several variables. We say reunification since we think that Weierstrass, Poincare, and others (in contrast to many of our students) did not consider them to be entirely separate subjects. Indeed, not only complex analysis in several variables, but also number theory, harmonic analysis, and other branches of mathematics, both pure and applied, have required a reconsidera tion of analytic continuation, ordinary differential equations in the complex domain, asymptotic analysis, iteration of holomorphic functions, and many other subjects from the classic theory of functions of one complex variable. This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written. |
Contents
II | vii |
III | xi |
IV | 6 |
V | 11 |
VI | 16 |
VII | 30 |
VIII | 44 |
IX | 54 |
XXXVII | 288 |
XXXVIII | 292 |
XXXIX | 306 |
XL | 340 |
XLI | 354 |
XLII | 380 |
XLIII | 401 |
XLIV | 421 |
X | 59 |
XI | 72 |
XII | 79 |
XIII | 86 |
XVI | 105 |
XVII | 116 |
XVIII | 126 |
XIX | 136 |
XX | 153 |
XXI | 166 |
XXII | 180 |
XXIII | 201 |
XXV | 209 |
XXVI | 216 |
XXVII | 223 |
XXVIII | 225 |
XXIX | 233 |
XXXII | 256 |
XXXIII | 266 |
XXXIV | 287 |
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Common terms and phrases
a₁ a₂ Algebraic analytic continuation assume B(zo b₁ biholomorphic map bounded compact set compact subset compute conclude connected component connected open set consider constant continuous function converges Corollary covering map defined Definition denote differential form Dirichlet series disk dx dy entire function Exercise finite follows formula function f half-plane harmonic function hence holomorphic function homeomorphism homotopic implies inequality injective integral isomorphism Jordan curve Lemma Let f lim sup linear log|z logr loop meromorphic function Moebius transformation N₁ N₂ neighborhood obtain open set open subset p₁ path pole polynomial PROOF Proposition prove r₁ Riemann surface satisfies sequence Show simply connected singularity space subharmonic function theorem Theory topological U₁ U₂ unique values vanishes vector verify w₁ w₂ z₁ zero ΖΕΩ ди
References to this book
Large-Time Behavior of Solutions of Linear Dispersive Equations Daniel B. Dix No preview available - 1997 |