06158cam a2200637Ii 45000010014000000030008000140050017000220060019000390070015000580080041000730400035001140200036001490200033001850200036002180200033002540200018002870200015003050200018003200200015003380200018003530200015003710200018003860200015004040350060004190350024004790500026005030720025005290720025005540720016005790820015005951000056006102450084006662500020007502640037007703000023008073360026008303370026008563380036008824900029009185050579009475050583015265050598021095050595027075050603033025201136039055880047050416500036050886500047051246500033051716500038052046500037052426500050052798560072053298560102054019990017055039780429275166FlBoTFG20260210180821.0m     o  d        cr cnu|||unuuu190419s2019    flu     ob    001 0 eng d  aOCoLC-PbengerdaepncOCoLC-P  a9780429275166q(electronic bk.)  a0429275161q(electronic bk.)  a9780415000352q(electronic bk.)  a0415000351q(electronic bk.)  z9780367222673  z0367222671  a9781000000351  a1000000354  a9781000013719  a1000013715  a9781000007183  a1000007189  a(OCoLC)1097665040z(OCoLC)1097959693z(OCoLC)1097985226  a(OCoLC-P)1097665040 4aQA331.7b.K732 2019eb 7aMATx0050002bisacsh 7aMATx0340002bisacsh 7aPBK2bicssc04a515/.92231 aKrantz, Steven G.q(Steven George),d1951-eauthor.10aComplex variables :ba physical approach with applications /cSteven G. Krantz.  aSecond edition. 1aBoca Raton :bCRC Press,c[2019]  a1 online resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier1 aTextbooks in mathematics0 aCover; 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 Applications8 a2.1 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; Exercises8 a3.2 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 Theorem8 a4.2.3 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 Function8 a5.1.7 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  aWeb 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.  aOCLC-licensed vendor bibliographic record. 0aFunctions of complex variables. 0aFunctions of complex variablesvTextbooks. 0aNumbers, ComplexvTextbooks. 0aMathematical analysisvTextbooks. 7aMATHEMATICS / Calculus.2bisacsh 7aMATHEMATICS / Mathematical Analysis.2bisacsh403Taylor & Francisuhttps://www.taylorfrancis.com/books/9780429275166423OCLC metadata license agreementuhttp://www.oclc.org/content/dam/oclc/forms/terms/vbrl-201703.pdf  c91867d91866