Complex Variables Demystified

by McMahon, David

• ISBN13: 9780071549202
• ISBN10: 007154920X
• Edition: 1st
• Format: Paperback
• Copyright: 2008-07-14
• Publisher: McGraw Hill

Summary

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

Preface	p. xi
Complex Numbers	p. 1
The Algebra of Complex Numbers	p. 2
Complex Variables	p. 4
Rules for the Complex Conjugate	p. 5
Pascal's Triangle	p. 9
Axioms Satisfied by the Complex Number System	p. 10
Properties of the Modulus	p. 12
The Polar Representation	p. 12
The nth Roots of Unity	p. 16
Summary	p. 19
Quiz	p. 19
Functions, Limits, and Continuity	p. 21
Complex Functions	p. 21
Plotting Complex Functions	p. 28
Multivalued Functions	p. 33
Limits of Complex Functions	p. 33
Limits Involving Infinity	p. 38
Continuity	p. 38
Summary	p. 40
Quiz	p. 40
The Derivative and Analytic Functions	p. 41
The Derivative Defined	p. 42
Leibniz Notation	p. 43
Rules for Differentiation	p. 45
Derivatives of Some Elementary Functions	p. 47
The Product and Quotient Rules	p. 48
The Cauchy-Riemann Equations	p. 51
The Polar Representation	p. 57
Some Consequences of the Cauchy-Riemann Equations	p. 59
Harmonic Functions	p. 61
The Reflection Principle	p. 63
Summary	p. 64
Quiz	p. 64
Elementary Functions	p. 65
Complex Polynomials	p. 65
The Complex Exponential	p. 70
Trigonometric Functions	p. 75
The Hyperbolic Functions	p. 78
Complex Exponents	p. 84
Derivatives of Some Elementary Functions	p. 85
Branches	p. 88
Summary	p. 89
Quiz	p. 89
Sequences and Series	p. 91
Sequences	p. 91
Infinite Series	p. 94
Convergence	p. 94
Convergence Tests	p. 96
Uniformly Converging Series	p. 97
Power Series	p. 97
Taylor and Maclaurin Series	p. 98
Theorems on Power Series	p. 98
Some Common Series	p. 100
Laurent Series	p. 109
Types of Singularities	p. 111
Entire Functions	p. 112
Meromorphic Functions	p. 112
Summary	p. 114
Quiz	p. 114
Complex Integration	p. 117
Complex Functions w(t)	p. 117
Properties of Complex Integrals	p. 119
Contours in the Complex Plane	p. 121
Complex Line Integrals	p. 124
The Cauchy-Goursat Theorem	p. 127
Summary	p. 133
Quiz	p. 134
Residue Theory	p. 135
Theorems Related to Cauchy's Integral Formula	p. 135
The Cauchy's Integral Formula as a Sampling Function	p. 143
Some Properties of Analytic Functions	p. 144
The Residue Theorem	p. 148
Evaluation of Real, Definite Integrals	p. 151
Integral of a Rational Function	p. 155
Summary	p. 161
Quiz	p. 161
More Complex Integration and the Laplace Transform	p. 163
Contour Integration Continued	p. 163
The Laplace Transform	p. 167
The Bromvich Inversion Integral	p. 179
Summary	p. 181
Quiz	p. 181
Mapping and Transformations	p. 183
Linear Transformations	p. 184
The Transformation z[superscript n]	p. 188
Conformal Mapping	p. 190
The Mapping 1/z	p. 190
Mapping of Infinite Strips	p. 192
Rules of Thumb	p. 194
Mobius Transformations	p. 195
Fixed Points	p. 201
Summary	p. 202
Quiz	p. 202
The Schwarz-Christoffel Transformation	p. 203
The Riemann Mapping Theorem	p. 203
The Schwarz-Christoffel Transformation	p. 204
Summary	p. 207
Quiz	p. 207
The Gamma and Zeta Functions	p. 209
The Gamma Function	p. 209
More Properties of the Gamma Function	p. 219
Contour Integral Representation and Stirling's Formula	p. 224
The Beta Function	p. 224
The Riemann Zeta Function	p. 225
Summary	p. 230
Quiz	p. 230
Boundary Value Problems	p. 231
Laplace's Equation and Harmonic Functions	p. 231
Solving Boundary Value Problems Using Conformal Mapping	p. 234
Green's Functions	p. 244
Summary	p. 247
Quiz	p. 247
Final Exam	p. 249
Quiz Solutions	p. 255
Final Exam Solutions	p. 261
Bibliography	p. 267
Index	p. 269
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