## Description

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

Unlike typical , 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 need the skills covered in this book for their daily work, Advanced Engineering Mathematics also makes an excellent reference for practicing and .

## Table of Contents

<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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>
<p></p>
<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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