06158cam a2200637Ii 4500 9780429275166 FlBoTFG 20260210180821.0 m o d cr cnu|||unuuu 190419s2019 flu ob 001 0 eng d OCoLC-P eng rda pn OCoLC-P 9780429275166 (electronic bk.) 0429275161 (electronic bk.) 9780415000352 (electronic bk.) 0415000351 (electronic bk.) 9780367222673 0367222671 9781000000351 1000000354 9781000013719 1000013715 9781000007183 1000007189 (OCoLC)1097665040 (OCoLC)1097959693 (OCoLC)1097985226 (OCoLC-P)1097665040 QA331.7 .K732 2019eb MAT 005000 bisacsh MAT 034000 bisacsh PBK bicssc 515/.9 23 Krantz, Steven G. (Steven George), 1951- author. Complex variables : a physical approach with applications / Steven G. Krantz. Second edition. Boca Raton : CRC Press, [2019] 1 online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier Textbooks in mathematics 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 Web Copy The idea of complex numbers dates back at least 300 years--to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers. This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications. Features: This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysis This book has an exceptionally large number of examples and a large number of figures. The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking. Incisive applications appear throughout the book. Partial differential equations are used as a unifying theme. OCLC-licensed vendor bibliographic record. Functions of complex variables. Functions of complex variables Textbooks. Numbers, Complex Textbooks. Mathematical analysis Textbooks. MATHEMATICS / Calculus. bisacsh MATHEMATICS / Mathematical Analysis. bisacsh Taylor & Francis https://www.taylorfrancis.com/books/9780429275166 OCLC metadata license agreement http://www.oclc.org/content/dam/oclc/forms/terms/vbrl-201703.pdf 91867 91866