http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Cover; Half Title; Title Page; Copyright Page; Dedication; Table of Contents; Preface to the Second Edition for the Instructor; Preface to the Second Edition for the Student; Preface to the First Edition; 1: Basic Ideas; 1.1 Complex Arithmetic; 1.1.1 The Real Numbers; 1.1.2 The Complex Numbers; 1.1.3 Complex Conjugate; Exercises; 1.2 Algebraic and Geometric Properties; 1.2.1 Modulus of a Complex Number; 1.2.2 The Topology of the Complex Plane; 1.2.3 The Complex Numbers as a Field; 1.2.4 The Fundamental Theorem of Algebra; Exercises; 2: The Exponential and Applications
2.1 The Exponential Function2.1.1 Laws of Exponentiation; 2.1.2 The Polar Form of a Complex Number; Exercises; 2.1.3 Roots of Complex Numbers; 2.1.4 The Argument of a Complex Number; 2.1.5 Fundamental Inequalities; Exercises; 3: Holomorphic and Harmonic Functions; 3.1 Holomorphic Functions; 3.1.1 Continuously Differentiable and Ck Functions; 3.1.2 The Cauchy-Riemann Equations; 3.1.3 Derivatives; 3.1.4 Definition of Holomorphic Function; 3.1.5 Examples of Holomorphic Functions; 3.1.6 The Complex Derivative; 3.1.7 Alternative Terminology for Holomorphic Functions; Exercises
3.2 Holomorphic and Harmonic Functions3.2.1 Harmonic Functions; 3.2.2 Holomorphic and Harmonic Functions; Exercises; 3.3 Complex Differentiability; 3.3.1 Conformality; Exercises; 4: The Cauchy Theory; 4.1 Real and Complex Line Integrals; 4.1.1 Curves; 4.1.2 Closed Curves; 4.1.3 Differentiable and Ck Curves; 4.1.4 Integrals on Curves; 4.1.5 The Fundamental Theorem of Calculus along Curves; 4.1.6 The Complex Line Integral; 4.1.7 Properties of Integrals; Exercises; 4.2 The Cauchy Integral Theorem; 4.2.1 The Cauchy Integral Theorem, Basic Form; 4.2.2 More General Forms of the Cauchy Theorem
4.2.3 Deformability of Curves4.2.4 Cauchy Integral Formula, Basic Form; 4.2.5 More General Versions of the Cauchy Formula; Exercises; 4.3 Variants of the Cauchy Formula; 4.4 The Limitations of the Cauchy Formula; Exercises; 5: Applications of the Cauchy Theory; 5.1 The Derivatives of a Holomorphic Function; 5.1.1 A Formula for the Derivative; 5.1.2 The Cauchy Estimates; 5.1.3 Entire Functions and Liouville's Theorem; 5.1.4 The Fundamental Theorem of Algebra; 5.1.5 Sequences of Holomorphic Functions and Their Derivatives; 5.1.6 The Power Series Representation of a Holomorphic Function
5.1.7 Table of Elementary Power SeriesExercises; 5.2 The Zeros of a Holomorphic Function; 5.2.1 The Zero Set of a Holomorphic Function; 5.2.2 Discrete Sets and Zero Sets; 5.2.3 Uniqueness of Analytic Continuation; Exercises; 6: Isolated Singularities; 6.1 Behavior Near an Isolated Singularity; 6.1.1 Isolated Singularities; 6.1.2 A Holomorphic Function on a Punctured Domain; 6.1.3 Classification of Singularities; 6.1.4 Removable Singularities, Poles, and Essential Singularities; 6.1.5 The Riemann Removable Singularities Theorem; 6.1.6 The Casorati-Weierstrass Theorem; 6.1.7 Concluding Remarks
Includes bibliographical references and index.
Textbooks in mathematics (Boca Raton, Fla.)