ADVANCED ENGINEERING MATHEMATICS MATLAB

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1 ADVANCED ENGINEERING MATHEMATICS WITH MATLAB THIRD EDITION Dean G. Duffy

2 Contents Dedication Contents Acknowledgments Author Introduction List of Definitions Chapter 1: Complex Variables 1.1 Complex Numbers 1.2 Finding Roots 1.3 The Derivative in the Complex Plane: The Cauchy-Riemann Equations

3 1.4 Line Integrals 1.5 The Cauchy-Goursat Theorem 1.6 Cauchy's Integral Formula 1.7 Taylor and Laurent Expansions and Singularities 1.8 Theory of Residues 1.9 Evaluation of Real Definite Integrals 1.10 Cauchy's Principal Value Integral r Chapter 2: First-Order Ordinary Differential Equations Classification of Differential Equations 2.2 Separation of Variables 2.3 Homogeneous Equations 2.4 Exact Equations 2.5 Linear Equations 2.6 Graphical Solutions 2.7 Numerical Methods Chapter 3: Higher-Order Ordinary Differential Equations Homogeneous Linear Equations with Constant Coefficients 120

4 3.2 Simple Harmonic Motion Damped Harmonic Motion ' Method of Undetermined Coefficients Forced Harmonic Motion Variation of Parameters Euler-Cauchy Equation Phase Diagrams Numerical Methods 170 Chapter 4: 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 222 Chapter 5: The Fourier Transform Fourier Transforms 5.2 Fourier Transforms Containing the Delta Function 5.3 Properties of Fourier Transforms

5 5.4 Inversion of Fourier Transforms Convolution ' Solution of Ordinary Differential Equations by Fourier Transforms 285 Chapter 6: 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 353 Chapter 7: The Z-Transform The Relationship of the Z-Transform to the Laplace Transform 1.2 Some Useful Properties 17.3 Inverse Z-Transforms 7.4 Solution of Difference Equations

6 7.5 Stability of Discrete-Time Systems f Chapter 8: The Hilbert Transform 8.1 Definition 8.2 Some Useful Properties 8.3 Analytic Signals 8.4 Causality: The Kramers-Kronig Relationship Chapter 9: The Sturm-Liouville Problem 9.1 Eigenvalues and Eigenfunctions 9.2 Orthogonality of Eigenfunctions 9.3 Expansion in Series of Eigenfunctions 9.4 A Singular Sturm-Liouville Problem: Legendre's Equation 9.5 Another Singular Sturm-Liouville Problem: Bessel's Equation 9.6 Finite Element Method Chapter 10: The Wave Equation 10.1 The Vibrating String

7 10.2 Initial Conditions: Cauchy Problem 502 f 10.3 Separation of Variables D'Alembert's Formula The Laplace Transform Method Numerical Solution of the Wave Equation 553 Chapter 11: 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 649 _ - Chapter 12: " Laplace's Equation 659 " " o * i 12.1 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 713

8 12.7 Numerical Solution of Laplace's Equation 12.8 Finite Element Solution of aplace's Equation Chapter 13: Green's Functions 13.1 What Is a Green's Function? 13.2 Ordinary Differential Equations 13.3 Joint Transform Method 13.4 Wave Equation 13.5 Heat Equation 13.6 Helmholtz's Equation Chapter 14: Vector Calculus x 14.1 Review 14.2 Divergence and Curl 14.3 Line Integrals 14.4 The Potential Function 14.5 Surface Integrals 14.6 Green's Lemma 14.7 Stokes' Theorem 14.8 Divergence Theorem

9 / ail o, 12 ' ' ain \ ; a2n Chapter 15: Linear Algebra 863 \ Oml ^m2 * ' ' ^mn / 15.1 Fundamentals of Linear Algebra Determinants Cramer's Rule Row Echelon Form and Gaussian Elimination Eigenvalues ; and Eigenvectors Systems of Linear Differential Equations Matrix Exponential 905 Chapter 16: Probability Review of Set Theory Classic Probability Discrete Random Variables Continuous Random Variables Mean and Variance Some Commonly Used Distributions Joint Distributions 956 Chapter 17: Random Processes Fundamental Concepts 973

10 17.2 Power Spectrum Differential Equations Forced by Random Forcing Two-State Markov Chains Birth and Death Processes Poisson Processes Random Walk 1024 Answers to the Odd-Numbered Problems 1037 Index 1067

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS DENNIS G. ZILL Loyola Marymount University MICHAEL R. CULLEN Loyola Marymount University PWS-KENT O I^7 3 PUBLISHING COMPANY E 9 U Boston CONTENTS Preface xiii Parti ORDINARY