What is included with this book?
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.
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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