Functions of One Complex Variable, Volume 1This 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 - I) 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. |
Contents
III | 11 |
Elementary Properties and Examples of Analytic Functions | 30 |
346 | 36 |
Copyright | |
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a₁ analytic continuation analytic function analytic on G assume bounded Cauchy sequence closed rectifiable curve compact subset complete analytic function complex numbers component constant contains continuous function convex Corollary curve in G Definition differentiable disk entire function equation Exercise f₁ fact finite number follows formula function element function f function on G ƒ and g ƒ is analytic G₁ gives harmonic function Hence homeomorphism homotopic implies infinite integer Lemma Let f Let ƒ Let G lim sup limit point meromorphic function metric space Möbius transformation one-one open set path plane point in G pole polynomial Proof properties Proposition prove R₁ R₂ rational function reader real number rectifiable curve region G removable singularity satisfies set G show that ƒ simply connected subharmonic subset of G Suppose f suppose that f topological space uniformly continuous