Complex Analysis Complex Analysis Complex Analysis: An Introduction to the Theory of Analytic Functions of One Can Introduction to the Theory of Analy

Revised
ISBN: 978-0-07-000657-7
Verlag: MCGRAW HILL BOOK CO

Buch, Englisch, Band 0007, 352 Seiten, Format (B × H): 162 mm x 238 mm, Gewicht: 644 g

Reihe: International Series in Pure &

A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals.

Autoren/Hrsg.

Ahlfors, Lars

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis

Inhaltsverzeichnis

Chapter 1: Complex Numbers1 The Algebra of Complex Numbers1.1Arithmetic Operations1.2Square Roots1.3Justification1.4Conjugation, Absolute Value1.5Inequalities2 The Geometric Representation of Complex Numbers2.1Geometric Addition and Multiplication2.2The Binomial Equation2.3Analytic Geometry2.4The Spherical RepresentationChapter 2: Complex Functions1 Introduction to the Concept of Analytic Function1.1Limits and Continuity1.2Analytic Functions1.3Polynomials1.4Rational Functions2 Elementary Theory of Power Series2.1Sequences2.2Series2.3Uniform Coverages2.4Power Series2.5 Abel's Limit Theorem3 The Exponential and Trigonometric Functions3.1 The Exponential3.2 The Trigonometric Functions3.3 The Periodicity3.4 The LogarithmChapter 3: Analytic Functions as Mappings1 Elementary Point Set Topology1.1Sets and Elements1.2Metric Spaces1.3Connectedness1.4Compactness1.5 Continuous Functions1.6 Topological Spaces2 Conformality2.1Arcs and Closed Curves2.2Analytic Functions in Regions2.3Conformal Mapping2.4Length and Area3 Linear Transformations3.1 The Linear Group3.2 The Cross Ratio3.3 Symmetry3.4 Oriented Circles3.5 Families of Circles4 Elementary Conformal Mappings4.1 The Use of Level Curves4.2 A Survey of Elementary Mappings4.3 Elementary Riemann SurfacesChapter 4: Complex Integration1 Fundamental Theorems1.1Line Integrals1.2Rectifiable Arcs1.3Line Integrals as Functions of Arcs1.4Cauchy's Theorem for a Rectangle1.5 Cauchy's Theorem in a Disk2 Cauchy's Integral Formula2.1The Index of a Point with Respect to a Closed Curve2.2The Integral Formula2.3Higher Derivatives3 Local Properties of Analytical Functions3.1 Removable Singularities. Taylor's Theorem3.2 Zeros and Poles3.3 The Local Mapping3.4 The Maximum Principle4 The General Form of Cauchy