**COMPLEX VARIABLES**

Complex Numbers

Finding Roots

The Derivative in the Complex Plane: The Cauchy–Riemann Equations

Line Integrals

Cauchy–Goursat Theorem

Cauchy’s Integral Formula

Taylor and Laurent Expansions and Singularities

Theory of Residues

Evaluation of Real Definite Integrals

Cauchy’s 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

Euler–Cauchy 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: Heaviside’s 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 Kramers–Kronig Relationship

**THE STURM–LIOUVILLE PROBLEM**

Eigenvalues and Eigenfunctions

Orthogonality of Eigenfunctions

Expansion in Series of Eigenfunctions

A Singular Sturm–Liouville Problem: Legendre’s Equation

Another Singular Sturm–Liouville Problem: Bessel’s Equation

Finite Element Method

**THE WAVE EQUATION**

The Vibrating String

Initial Conditions: Cauchy Problem

Separation of Variables

D’Alembert’s 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

**LAPLACE’S EQUATION**

Derivation of Laplace’s Equation

Boundary Conditions

Separation of Variables

The Solution of Laplace’s Equation on the Upper Half-Plane

Poisson’s Equation on a Rectangle

The Laplace Transform Method

Numerical Solution of Laplace’s Equation

Finite Element Solution of Laplace’s Equation

**GREEN’S FUNCTIONS **

What Is a Green’s Function?

Ordinary Differential Equations

Joint Transform Method

Wave Equation

Heat Equation

Helmholtz’s Equation

**VECTOR CALCULUS**

Review

Divergence and Curl

Line Integrals

The Potential Function

Surface Integrals

Green’s Lemma

Stokes’ Theorem

Divergence Theorem

**LINEAR ALGEBRA**

Fundamentals of Linear Algebra

Determinants

Cramer’s 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**

## No Comments

Feedback:

Leave your comments here!Your opinions and comments would be greatly appreciated.

If you have comments or questions we've added this section so that we might have a dialogue with you.