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The Derivative and Analytic Functions The Derivative De ned Leibniz Notation Rules for Differentiation Derivatives of Some Elementary Functions The Product and Quotient Rules The Cauchy-Riemann Equations The Polar Representation Some Consequences of the Cauchy-Riemann Equations Harmonic Functions The Re ection Principle Summary Quiz Elementary Functions Complex Polynomials The Complex Exponential Trigonometric Functions The Hyperbolic Functions Complex Exponents Derivatives of Some Elementary Functions Branches Summary Quiz Sequences and Series Sequences In nite Series Convergence Convergence Tests Uniformly Converging Series Power Series Taylor and Maclaurin Series Theorems on Power Series Some Common Series 41 42 43 45 47 48 51 57 59 61 63 64 64 65 65 70 75 78 84 85 88 89 89 91 91 94 94 96 97 97 98 98 100
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Contents
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Laurent Series Types of Singularities Entire Functions Meromorphic Functions Summary Quiz CHAPTER 6 Complex Integration Complex Functions w(t) Properties of Complex Integrals Contours in the Complex Plane Complex Line Integrals The Cauchy-Goursat Theorem Summary Quiz Residue Theory Theorems Related to Cauchy s Integral Formula The Cauchy s Integral Formula as a Sampling Function Some Properties of Analytic Functions The Residue Theorem Evaluation of Real, De nite Integrals Integral of a Rational Function Summary Quiz More Complex Integration and the Laplace Transform Contour Integration Continued The Laplace Transform The Bromvich Inversion Integral Summary Quiz Mapping and Transformations Linear Transformations The Transformation zn Conformal Mapping
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109 111 112 112 114 114 117 117 119 121 124 127 133 134 135 135 143 144 148 151 155 161 161 163 163 167 179 181 181 183 184 188 190
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The Mapping 1/z Mapping of In nite Strips Rules of Thumb M bius Transformations Fixed Points Summary Quiz 190 192 194 195 201 202 202 203 203 204 207 207 209 209 219 224 224 225 230 230 231 231 234 244 247 247 249 255 261 267 269
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The Schwarz-Christoffel Transformation The Riemann Mapping Theorem The Schwarz-Christoffel Transformation Summary Quiz The Gamma and Zeta Functions The Gamma Function More Properties of the Gamma Function Contour Integral Representation and Stirling s Formula The Beta Function The Riemann Zeta Function Summary Quiz Boundary Value Problems Laplace s Equation and Harmonic Functions Solving Boundary Value Problems Using Conformal Mapping Green s Functions Summary Quiz Final Exam Quiz Solutions Final Exam Solutions Bibliography Index
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Complex variables, and its more advanced version, complex analysis, is one of the most fascinating areas in pure and applied mathematics It all started when mathematicians were mystified by equations that could only be solved if you could take the square roots of negative numbers This seemed bizarre, and back then nobody could imagine that something as strange as this could have any application in the real world Thus the term imaginary number was born and the area seemed so odd it became known as complex These terms have stuck around even though the theory of complex variables has found a home as a fundamental part of mathematics and has a wide range of physical applications In mathematics, it turns out that complex variables are actually an extension of the real variables A student planning on becoming a professional pure or applied mathematician should definitely have a thorough grasp of complex analysis Perhaps the most surprising thing about complex variables is the wide range of applications it touches in physics and engineering In many of these applications, complex variables proves to be a useful tool For example, because of Euler s identity, a formula we use over and over again in this book, electromagnetic fields are easier to deal with using complex variables Other areas where complex variables plays a role include fluid dynamics, the study of temperature, electrostatics, and in the evaluation of many real integrals of functions of a real variable In quantum theory, we meet the most surprising revelation about complex variables It turns out they are not so imaginary at all Instead, they appear to be as real as real numbers and even play a fundamental role in the working of physical systems at the microscopic level In the limited space of this book, we won t be able to cover the physical applications of complex variables Our purpose here is to build a solid foundation to get you started on the subject This book is filled with a large number of solved
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