This text/reference covers essential areas of engineering mathematics involving single, multiple, and complex variations. Taken as a whole, this book provides a succinct, carefully organized guide for mastering engineering mathematics.

Unlike typical textbooks, Advanced Engineering Mathematics begins with a thorough exploration of complex variables because they provide powerful techniques for understanding topics, such as Fourier, Laplace and z-transforms, introduced later in the text. The book contains a wealth of examples, both classic problems used to illustrate concepts, and interesting real-life examples from scientific literature.

Ideal for a two-semester course on advanced engineering mathematics, Advanced Engineering Mathematics is concise and well-organized, unlike the long, detailed texts used to teach this subject. Since almost every engineer and many scientists need the skills covered in this book for their daily work, Advanced Engineering Mathematics also makes an excellent reference for practicing engineers and scientists.

**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**

REVIEW