<p><strong>COMPLEX VARIABLES</strong><br>Complex Numbers<br>Finding Roots<br>The Derivative in the Complex Plane: The Cauchy–Riemann Equations<br>Line Integrals<br>Cauchy–Goursat Theorem<br>Cauchy’s Integral Formula<br>Taylor and Laurent Expansions and Singularities<br>Theory of Residues<br>Evaluation of Real Definite Integrals<br>Cauchy’s Principal Value Integral</p>
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<p><strong>FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS</strong><br>Classification of Differential Equations<br>Separation of Variables<br>Homogeneous Equations<br>Exact Equations<br>Linear Equations<br>Graphical Solutions<br>Numerical Methods</p>
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<p><strong>HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS</strong><br>Homogeneous Linear Equations with Constant Coefficients<br>Simple Harmonic Motion<br>Damped Harmonic Motion<br>Method of Undetermined Coefficients<br>Forced Harmonic Motion<br>Variation of Parameters<br>Euler–Cauchy Equation<br>Phase Diagrams<br>Numerical Methods</p>
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<p><strong>FOURIER SERIES</strong><br>Fourier Series<br>Properties of Fourier Series<br>Half-Range Expansions<br>Fourier Series with Phase Angles<br>Complex Fourier Series<br>The Use of Fourier Series in the Solution of Ordinary Differential Equations<br>Finite Fourier Series</p>
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<p><strong>THE FOURIER TRANSFORM</strong><br>Fourier Transforms<br>Fourier Transforms Containing the Delta Function<br>Properties of Fourier Transforms<br>Inversion of Fourier Transforms<br>Convolution<br>Solution of Ordinary Differential Equations by Fourier Transforms</p>
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<p><strong>THE LAPLACE TRANSFORM</strong><br>Definition and Elementary Properties<br>The Heaviside Step and Dirac Delta Functions<br>Some Useful Theorems<br>The Laplace Transform of a Periodic Function<br>Inversion by Partial Fractions: Heaviside’s Expansion Theorem<br>Convolution<br>Integral Equations<br>Solution of Linear Differential Equations with Constant Coefficients<br>Inversion by Contour Integration</p>
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<p><strong>THE Z-TRANSFORM</strong><br>The Relationship of the Z-Transform to the Laplace Transform<br>Some Useful Properties<br>Inverse Z-Transforms<br>Solution of Difference Equations<br>Stability of Discrete-Time Systems</p>
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<p><strong>THE HILBERT TRANSFORM</strong><br>Definition<br>Some Useful Properties<br>Analytic Signals<br>Causality: The Kramers–Kronig Relationship</p>
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<p><strong>THE STURM–LIOUVILLE PROBLEM</strong><br>Eigenvalues and Eigenfunctions<br>Orthogonality of Eigenfunctions<br>Expansion in Series of Eigenfunctions<br>A Singular Sturm–Liouville Problem: Legendre’s Equation<br>Another Singular Sturm–Liouville Problem: Bessel’s Equation<br>Finite Element Method</p>
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<p><strong>THE WAVE EQUATION</strong><br>The Vibrating String<br>Initial Conditions: Cauchy Problem<br>Separation of Variables<br>D’Alembert’s Formula<br>The Laplace Transform Method<br>Numerical Solution of the Wave Equation</p>
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<p><strong>THE HEAT EQUATION</strong><br>Derivation of the Heat Equation<br>Initial and Boundary Conditions<br>Separation of Variables<br>The Laplace Transform Method<br>The Fourier Transform Method<br>The Superposition Integral<br>Numerical Solution of the Heat Equation</p>
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<p><strong>LAPLACE’S EQUATION</strong><br>Derivation of Laplace’s Equation<br>Boundary Conditions<br>Separation of Variables<br>The Solution of Laplace’s Equation on the Upper Half-Plane<br>Poisson’s Equation on a Rectangle<br>The Laplace Transform Method<br>Numerical Solution of Laplace’s Equation<br>Finite Element Solution of Laplace’s Equation</p>
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<p><strong>GREEN’S FUNCTIONS <br></strong>What Is a Green’s Function? <br>Ordinary Differential Equations <br>Joint Transform Method <br>Wave Equation<br>Heat Equation <br>Helmholtz’s Equation</p>
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<p><strong>VECTOR CALCULUS</strong><br>Review<br>Divergence and Curl<br>Line Integrals<br>The Potential Function<br>Surface Integrals<br>Green’s Lemma<br>Stokes’ Theorem<br>Divergence Theorem</p>
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<p><strong>LINEAR ALGEBRA</strong><br>Fundamentals of Linear Algebra<br>Determinants<br>Cramer’s Rule<br>Row Echelon Form and Gaussian Elimination<br>Eigenvalues and Eigenvectors<br>Systems of Linear Differential Equations<br>Matrix Exponential</p>
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<p><strong>PROBABILITY</strong><br>Review of Set Theory <br>Classic Probability <br>Discrete Random Variables <br>Continuous Random Variables <br>Mean and Variance <br>Some Commonly Used Distributions <br>Joint Distributions</p>
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<p><strong>RANDOM PROCESSES</strong> <br>Fundamental Concepts<br>Power Spectrum <br>Differential Equations Forced by Random Forcing <br>Two-State Markov Chains <br>Birth and Death Processes <br>Poisson Processes <br>Random Walk</p>
<p><strong>ANSWERS TO THE ODD-NUMBERED PROBLEMS</strong></p>
<p><strong>INDEX</strong></p>
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