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