ADVANCED ENGINEERING MATHEMATICS WITH MATLAB THIRD EDITION Dean G. Duffy
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
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 19 25 28 32 40 45 54 Chapter 2: First-Order Ordinary Differential Equations 63 2.1 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 63 67 81 82 85 98 101 Chapter 3: Higher-Order Ordinary Differential Equations 115 3.1 Homogeneous Linear Equations with Constant Coefficients 120
3.2 Simple Harmonic Motion 129 3.3 Damped Harmonic Motion ' 134 3.4 Method of Undetermined Coefficients 139 3.5 Forced Harmonic Motion 145 3.6 Variation of Parameters 154 3.7 Euler-Cauchy Equation 160 3.8 Phase Diagrams 164 3.9 Numerical Methods 170 Chapter 4: Fourier Series 177 4.1 Fourier Series 177 4.2 Properties of Fourier Series 191 4.3 Half-Range Expansions 199 4.4 Fourier Series with Phase Angles 206 4.5 Complex Fourier Series 208 4.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations 213 4.7 Finite Fourier Series 222 Chapter 5: The Fourier Transform 239 5.1 Fourier Transforms 5.2 Fourier Transforms Containing the Delta Function 5.3 Properties of Fourier Transforms 239 250 253
5.4 Inversion of Fourier Transforms 267 5.5 Convolution ' 282 5.6 Solution of Ordinary Differential Equations by Fourier Transforms 285 Chapter 6: The Laplace Transform 289 6.1 Definition and Elementary Properties 289 6.2 The Heaviside Step and Dirac Delta Functions 297 6.3 Some Useful Theorems 303 6.4 The Laplace Transform of a Periodic Function 312 6.5 Inversion by Partial Fractions: Heaviside's Expansion Theorem 315 6.6 Convolution 323 6.7 Integral Equations 328 6.8 Solution of Linear Differential Equations with Constant Coefficients 334 6.9 Inversion by Contour Integration 353 Chapter 7: The Z-Transform 363 7.1 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 364 371 379 389
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
10.2 Initial Conditions: Cauchy Problem 502 f 10.3 Separation of Variables 503 10.4 D'Alembert's Formula 524 10.5 The Laplace Transform Method 532 10.6 Numerical Solution of the Wave Equation 553 Chapter 11: The Heat Equation 565 11.1 Derivation of the Heat Equation 566 11.2 Initial and Boundary Conditions 567 11.3 Separation of Variables 568 11.4 The Laplace Transform Method 612 11.5 The Fourier Transform Method 629 11.6 The Superposition Integral 636 11.7 Numerical Solution of the Heat Equation 649 _ - Chapter 12: " Laplace's Equation 659 " " o * i 12.1 Derivation of Laplace's Equation 660 12.2 Boundary Conditions 662 12.3 Separation of Variables 663 12.4 The Solution of Laplace's Equation on the Upper Half-Plane 707 12.5 Poisson's Equation on a Rectangle 709 12.6 The Laplace Transform Method 713
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
/ ail o, 12 ' ' ain \ 21 022 ; a2n Chapter 15: Linear Algebra 863 \ Oml ^m2 * ' ' ^mn / 15.1 Fundamentals of Linear Algebra 863 15.2 Determinants 871 15.3 Cramer's Rule 876 15.4 Row Echelon Form and Gaussian Elimination 879 15.5 Eigenvalues ; and Eigenvectors 890 15.6 Systems of Linear Differential Equations 899 15.7 Matrix Exponential 905 Chapter 16: Probability 913 16.1 Review of Set Theory 914 16.2 Classic Probability 916 16.3 Discrete Random Variables 928 16.4 Continuous Random Variables 934 16.5 Mean and Variance 942 16.6 Some Commonly Used Distributions 947 16.7 Joint Distributions 956 Chapter 17: Random Processes 969 17.1 Fundamental Concepts 973
17.2 Power Spectrum 980 17.3 Differential Equations Forced by Random Forcing 984 17.4 Two-State Markov Chains 993 17.5 Birth and Death Processes 1003 17.6 Poisson Processes 1017 17.7 Random Walk 1024 Answers to the Odd-Numbered Problems 1037 Index 1067