Complex Variables Demystified

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  • Edition: 1st
  • Format: Paperback
  • Copyright: 2008-07-14
  • Publisher: McGraw-Hill Education
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Taking the Complication out of Complex Variables Complex variables is an essential mathematical and engineering field, playing a crucial role in signal processing, electromagnetics, image analysis, differential equations, mathematical modeling, fluid flow, astrophysics, and modern analytical science. Written in the bestselling Demystified format, this user-friendly guide offers a straightforward way for you to comprehend this complex topic. Hundreds of examples and worked equations make it easy to understand the material, and end-of-chapter questions and a final exam help reinforce learning.

Author Biography

David McMahon has worked for several years asa physicist and researcher at Sandia National Laboratories. He is the author of Linear Algebra Demystified, Quantum Mechanics Demystified, Relativity Demystified, and MATLAB Demystified, among other successful titles.

Table of Contents

Prefacep. xi
Complex Numbersp. 1
The Algebra of Complex Numbersp. 2
Complex Variablesp. 4
Rules for the Complex Conjugatep. 5
Pascal's Trianglep. 9
Axioms Satisfied by the Complex Number Systemp. 10
Properties of the Modulusp. 12
The Polar Representationp. 12
The nth Roots of Unityp. 16
Summaryp. 19
Quizp. 19
Functions, Limits, and Continuityp. 21
Complex Functionsp. 21
Plotting Complex Functionsp. 28
Multivalued Functionsp. 33
Limits of Complex Functionsp. 33
Limits Involving Infinityp. 38
Continuityp. 38
Summaryp. 40
Quizp. 40
The Derivative and Analytic Functionsp. 41
The Derivative Definedp. 42
Leibniz Notationp. 43
Rules for Differentiationp. 45
Derivatives of Some Elementary Functionsp. 47
The Product and Quotient Rulesp. 48
The Cauchy-Riemann Equationsp. 51
The Polar Representationp. 57
Some Consequences of the Cauchy-Riemann Equationsp. 59
Harmonic Functionsp. 61
The Reflection Principlep. 63
Summaryp. 64
Quizp. 64
Elementary Functionsp. 65
Complex Polynomialsp. 65
The Complex Exponentialp. 70
Trigonometric Functionsp. 75
The Hyperbolic Functionsp. 78
Complex Exponentsp. 84
Derivatives of Some Elementary Functionsp. 85
Branchesp. 88
Summaryp. 89
Quizp. 89
Sequences and Seriesp. 91
Sequencesp. 91
Infinite Seriesp. 94
Convergencep. 94
Convergence Testsp. 96
Uniformly Converging Seriesp. 97
Power Seriesp. 97
Taylor and Maclaurin Seriesp. 98
Theorems on Power Seriesp. 98
Some Common Seriesp. 100
Laurent Seriesp. 109
Types of Singularitiesp. 111
Entire Functionsp. 112
Meromorphic Functionsp. 112
Summaryp. 114
Quizp. 114
Complex Integrationp. 117
Complex Functions w(t)p. 117
Properties of Complex Integralsp. 119
Contours in the Complex Planep. 121
Complex Line Integralsp. 124
The Cauchy-Goursat Theoremp. 127
Summaryp. 133
Quizp. 134
Residue Theoryp. 135
Theorems Related to Cauchy's Integral Formulap. 135
The Cauchy's Integral Formula as a Sampling Functionp. 143
Some Properties of Analytic Functionsp. 144
The Residue Theoremp. 148
Evaluation of Real, Definite Integralsp. 151
Integral of a Rational Functionp. 155
Summaryp. 161
Quizp. 161
More Complex Integration and the Laplace Transformp. 163
Contour Integration Continuedp. 163
The Laplace Transformp. 167
The Bromvich Inversion Integralp. 179
Summaryp. 181
Quizp. 181
Mapping and Transformationsp. 183
Linear Transformationsp. 184
The Transformation z[superscript n]p. 188
Conformal Mappingp. 190
The Mapping 1/zp. 190
Mapping of Infinite Stripsp. 192
Rules of Thumbp. 194
Mobius Transformationsp. 195
Fixed Pointsp. 201
Summaryp. 202
Quizp. 202
The Schwarz-Christoffel Transformationp. 203
The Riemann Mapping Theoremp. 203
The Schwarz-Christoffel Transformationp. 204
Summaryp. 207
Quizp. 207
The Gamma and Zeta Functionsp. 209
The Gamma Functionp. 209
More Properties of the Gamma Functionp. 219
Contour Integral Representation and Stirling's Formulap. 224
The Beta Functionp. 224
The Riemann Zeta Functionp. 225
Summaryp. 230
Quizp. 230
Boundary Value Problemsp. 231
Laplace's Equation and Harmonic Functionsp. 231
Solving Boundary Value Problems Using Conformal Mappingp. 234
Green's Functionsp. 244
Summaryp. 247
Quizp. 247
Final Examp. 249
Quiz Solutionsp. 255
Final Exam Solutionsp. 261
Bibliographyp. 267
Indexp. 269
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