Details

Complex Analysis


Complex Analysis

A Modern First Course in Function Theory
1. Aufl.

von: Jerry R. Muir

72,99 €

Verlag: Wiley
Format: PDF
Veröffentl.: 06.05.2015
ISBN/EAN: 9781118956397
Sprache: englisch
Anzahl Seiten: 280

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Beschreibungen

<p><b>A thorough introduction to the theory of complex functions emphasizing the beauty, power, and counterintuitive nature of the subject</b></p> <p>Written with a reader-friendly approach, <i>Complex Analysis: A Modern First Course in Function Theory </i>features a self-contained, concise development of the fundamental principles of complex analysis. After laying groundwork on complex numbers and the calculus and geometric mapping properties of functions of a complex variable, the author uses power series as a unifying theme to define and study the many rich and occasionally surprising properties of analytic functions, including the Cauchy theory and residue theorem. The book concludes with a treatment of harmonic functions and an epilogue on the Riemann mapping theorem.</p> <p>Thoroughly classroom tested at multiple universities, <i>Complex Analysis: A Modern First Course in Function Theory </i>features:</p> <ul> <li>Plentiful exercises, both computational and theoretical, of varying levels of difficulty, including several that could be used for student projects</li> <li>Numerous figures to illustrate geometric concepts and constructions used in proofs</li> <li>Remarks at the conclusion of each section that place the main concepts in context, compare and contrast results with the calculus of real functions, and provide historical notes</li> <li>Appendices on the basics of sets and functions and a handful of useful results from advanced calculus</li> </ul> Appropriate for students majoring in pure or applied mathematics as well as physics or engineering, <i>Complex Analysis: A Modern First Course in Function Theory </i>is an ideal textbook for a one-semester course in complex analysis for those with a strong foundation in multivariable calculus. The logically complete book also serves as a key reference for mathematicians, physicists, and engineers and is an excellent source for anyone interested in independently learning or reviewing the beautiful subject of complex analysis.<br /> <p> </p> <p> </p>
<p>Preface ix</p> <p><b>1 The Complex Numbers 1</b></p> <p>1.1 Why? 1</p> <p>1.2 The Algebra of Complex Numbers 3</p> <p>1.3 The Geometry of the Complex Plane 7</p> <p>1.4 The Topology of the Complex Plane 9</p> <p>1.5 The Extended Complex Plane 16</p> <p>1.6 Complex Sequences 18</p> <p>1.7 Complex Series 24</p> <p><b>2 Complex Functions and Mappings 29</b></p> <p>2.1 Continuous Functions 29</p> <p>2.2 Uniform Convergence 34</p> <p>2.3 Power Series 38</p> <p>2.4 Elementary Functions and Euler’s Formula 43</p> <p>2.5 Continuous Functions as Mappings 50</p> <p>2.6 Linear Fractional Transformations 53</p> <p>2.7 Derivatives 64</p> <p>2.8 The Calculus of Real Variable Functions 70</p> <p>2.9 Contour Integrals 75</p> <p><b>3 Analytic Functions 87</b></p> <p>3.1 The Principle of Analyticity 87</p> <p>3.2 Differentiable Functions are Analytic 89</p> <p>3.3 Consequences of Goursat’s Theorem 100</p> <p>3.4 The Zeros of Analytic Functions 104</p> <p>3.5 The Open Mapping Theorem and Maximum Principle 108</p> <p>3.6 The Cauchy–Riemann Equations 113</p> <p>3.7 Conformal Mapping and Local Univalence 117</p> <p><b>4 Cauchy’s Integral Theory 127</b></p> <p>4.1 The Index of a Closed Contour 127</p> <p>4.2 The Cauchy Integral Formula 133</p> <p>4.3 Cauchy’s Theorem 139</p> <p><b>5 The Residue Theorem 145</b></p> <p>5.1 Laurent Series 145</p> <p>5.2 Classification of Singularities 152</p> <p>5.3 Residues 158</p> <p>5.4 Evaluation of Real Integrals 165</p> <p>5.5 The Laplace Transform 174</p> <p><b>6 Harmonic Functions and Fourier Series 183</b></p> <p>6.1 Harmonic Functions 183</p> <p>6.2 The Poisson Integral Formula 191</p> <p>6.3 Further Connections to Analytic Functions 201</p> <p>6.4 Fourier Series 210</p> <p>Epilogue 227</p> <p>A Sets and Functions 239</p> <p>B Topics from Advanced Calculus 247</p> <p>References 255</p> <p>Index 257</p>
<p>"The textbook is appropriate for students and can serve as a key reference for anyone interested in learning or reviewing the theory of complex functions of a complex variable." (<i>Zentralblatt MATH</i>, 2016)</p>
<p><b>Jerry R. Muir, Jr., PhD,</b> is Professor of Mathematics at The University of Scranton. He has authored over one dozen research articles in complex-flavored analysis, primarily on geometric function theory in several complex variables.
<p><b>A thorough introduction to the theory of complex functions emphasizing the beauty, power, and counterintuitive nature of the subject</b> <p>Written with a reader-friendly approach, <i>Complex Analysis: A Modern First Course in Function Theory</i> features a self-contained, concise development of the fundamental principles of complex analysis. After laying groundwork on complex numbers and the calculus and geometric mapping properties of functions of a complex variable, the author uses power series as a unifying theme to define and study the many rich and occasionally surprising properties of analytic functions, including the Cauchy theory and residue theorem. The book concludes with a treatment of harmonic functions and an epilogue on the Riemann mapping theorem. <p>Thoroughly classroom tested at multiple universities, <i>Complex Analysis: A Modern First Course in Function Theory</i> features: <ul> <li>Plentiful exercises, both computational and theoretical, of varying levels of difficulty, including several that could be used for student projects</li> <li>Numerous figures to illustrate geometric concepts and constructions used in proofs</li> <li>Remarks at the conclusion of each section that place the main concepts in context, compare and contrast results with the calculus of real functions, and provide historical notes</li> <li>Appendices on the basics of sets and functions and a handful of useful results from advanced calculus</li> </ul> <p>Appropriate for students majoring in pure or applied mathematics as well as physics or engineering, <i>Complex Analysis: A Modern First Course in Function Theory</i> is an ideal textbook for a one-semester course in complex analysis for those with a strong foundation in multivariable calculus. The logically complete book also serves as a key reference for mathematicians, physicists, and engineers and is an excellent source for anyone interested in independently learning or reviewing the beautiful subject of complex analysis.

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