This text/reference covers essential areas of involving single, multiple, and complex variations. Taken as a whole, this 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 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
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
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
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

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
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

TO THE ODD-NUMBERED PROBLEMS

INDEX

Title: Advanced Engineering Mathematics with MATLAB
Author: Dean G. Duffy
Edition: 1st Edition
ISBN: 9781439816240
Type: eBook
Language: English
Advanced Mathematics
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