Lectures On The Theory Of Functions Of A Complex Variable - Info and Reading Options

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<p>In this post we will see the book <em>Lectures on the Theory of Functions of a Complex Variable</em> by Yu. V. Sidorov, <em>M. V. Fedoryuk</em>, <em>M. I. Shabunin.</em><br /></p><h2>About the book<br /></h2><blockquote><p>This book is based on more than ten years experience in teaching the theory of functions of a complex variable at the Moscow Physics and Technology Institute. It is a textbook for students of universities and institutes of technology with an advanced mathematical program.<br />We believe that it can also be used for independent study.<br />We have stressed the methods of the theory that are often used in applied sciences. These methods include series expansions, conformal mapping, application of the theory of residues to evaluating definite integrals, and asymptotic methods. The material is structured in a way that will give the reader the maximum assistance in mastering the basics of the theory. To this end we have provided a wide range of worked-out examples. We hope that these will help the reader<br />acquire a deeper understanding of the theory and experience in<br />problem solving.</p></blockquote><p>The book was translated from Russian by Eugene Yankovsky and was first published by Mir in 1985.</p><br /><h2>Contents<br /></h2><p>Preface 5</p><p><br /><strong>Chapter I Introduction 9</strong><br /></p><p>1 Complex Numbers 9<br />2 Sequences and Series of Complex Numbers 20<br />3 Curves and Domains in the Complex Plane 25<br />4 Continuous Functions of a Complex Variable 36<br />5 Integrating Functions of a Complex Variable 45<br />6 The Function arg z 51</p><p><br /><strong>Chapter II Regular Functions 59</strong><br /></p><p>7 Differentiable Functions. The Cauchy-Riemann Equations 59<br />8 The Geometric Interpretation of the Derivative 66<br />9 Cauchy's Integral Theorem 76 .--<br />10 Cauchy's Integral Formula 84<br />11 Power Series 87<br />12 Properties of Regular Functions 90<br />13 The Inverse Function 102<br />14 The Uniqueness Theorem 108<br />15 Analytic Continuation 110<br />16 Integrals Depending on a Parameter 112</p><p><br /><strong>Chapter III The Laurent Series. Isolated Singular Points of a Single-Valued Functions 123</strong></p><p><br />17 The Laurent Series 123<br />18 Isolated Singular Points of Single- Valued Functions 128<br />19 Liouville's Theorem 138</p><p><br /><strong>Chapter IV Multiple-Valued Analytic Functions 141</strong><br /></p><p>20 The Concept of an Analytic Function 141<br />21 The Function In z 147<br />22 The Power Function. Branch Points of Analytic Functions 155<br />23 The Primitive of an Analytic Function. Inverse Trigonometric<br />Functions 166<br />24 Regular Branches of Analytic Functions 170<br />25 Singular Boundary Points 189<br />26 Singular Points of Analytic Functions. The Concept of a Riemann<br />Surface 194<br />27 Analytic Theory of Linear Second-Order Ordinary Differential<br />Equations 204</p><p><strong>Chapter V Residues and Their Applications 220</strong></p><p><br />28 Residue Theorems 220<br />29 Use of Residues for Evaluating Definite Integrals 230<br />30 The Argument Principle and Rouche's Theorem 255<br />31 The Partial-Fraction Expansion of Meromorphic Functions 260</p><p><br /><strong>Chapter VI Conformal Mapping 270</strong><br /></p><p>32 Local Properties of Mappings Performed by Regular Functions 270<br />33 General Properties of Conformal Mappings 276<br />34 The Linear-Fractional Function 282<br />35 Conformal Mapping Performed by Elementary Functions 291<br />36 The Riemann-Schwarz Symmetry Principle 315<br />37 The Schwarz-Cristoffel Transformation Formula 326<br />38 The Dirichlet Problem 339<br />39 Vector Fields in a Plane 354<br />40 Some Physical Problems from Vector Field Theory 363</p><p><br /><strong>Chapter VII Simple Asymptotic Methods 371</strong><br /></p><p>41 Some Asymptotic Estimates 371<br />42 Asymptotic Expansions 389<br />43 Laplace's Method 396<br />44 The Method of Stationary Phase 409<br />45 The Saddle-Point Method '•18<br />46 Laplace's Method of Contour Integration 434<br /></p><p><strong>Chapter VIII Operational Calculus 446</strong><br /></p><p>47 Basic Properties of the Laplace Transformation 446<br />48 Reconstructing Object Function from Result Function 454<br />49 Solving Linear Differential Equations via the Laplace Transformation<br />468<br />50 String Vibrations from Instantaneous Shock 476</p><p><br />Selected Bibliography 486<br />Name Index 488<br />Subject Index 489</p><p><br /></p>

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