Advanced engineering mathematics with MATLAB / Dean G. Duffy

Contents Dedication Contents Acknowledgments Author Introduction List of Definitions chapter 1 Complex Variables 1 1.1. Complex Numbers 1 1.2. Finding Roots 6 1.3. The Derivative in the Complex Plane: The Cauchy-Riemann Equations 9 1.4. Line Integrals 19 1.5. The Cauchy-Goursat Theorem 25 1.6. Cauchy's Integral Formula 28 1.7. Taylor and Laurent Expansions and Singularities 32 1.8. Theory of Residues 40 1.9. Evaluation of Real Definite Integrals 45 1.10. Cauchy's Principal Value Integral 54 chapter 2 First-Order Ordinary Differential Equations 63 2.1. Classification of Differential Equations 63 2.2. Separation of Variables 67 2.3. Homogeneous Equations 81 2.4. Exact Equations 82 2.5. Linear Equations 85 2.6. Graphical Solutions 98 2.7. Numerical Methods 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 Transforms 239 5.1. Fourier Transforms 239 5.2. Fourier Transforms Containing the Delta Function 250 5.3. Properties of Fourier Transforms 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 364 7.2. Some Useful Properties 371 7.3. Inverse Z-Transforms 379 7.4. Solution of Difference Equations 389 7.5. Stability of Discrete-Time Systems 398 chapter 8 The Hilbert Transform 405 8.1. Definition 405 8.2. Some Useful Properties 416 8.3. Analytic Signals 423 8.4. Causality: The Kramers-Kronig Relationship 426 chapter 9 The Sturm-Liouville Problem 431 9.1. Eigenvalues and Eigenfunctions 431 9.2. Orthogonality of Eigenfunctions 442 9.3. Expansion in Series of Eigenfunctions 446 9.4. A Singular Sturm-Liouville Problem: Legendre's Equation 451 9.5. Another Singular Sturm-Liouville Problem: Bessel's Equation 468 9.6. Finite Element Method 490 chapter 10 The Wave Equation 499 10.1. The Vibrating String 500 10.2. Initial Conditions: Cauchy Problem 502 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 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 716 12.8. Finite Element Solution of Laplace's Equation 722 chapter 13 Green's Functions 731 13.1. What Is a Green's Function? 731 13.2. Ordinary Differential Equations 738 13.3. Joint Transform Method 762 13.4. Wave Equation 766 13.5. Heat Equation 777 13.6. Helmholtz's Equation 789 chapter 14 Vector Calculus 813 14.1. Review 813 14.2. Divergence and Curl 822 14.3. Line Integrals 827 14.4. The Potential Function 832 14.5. Surface Integrals 834 14.6. Green's Lemma 842 14.7. Stokes' Theorem 846 14.8. Divergence Theorem 853 chapter 15 Linear Algebra 863 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

"Resoundingly popular in its first edition, Dean Duffy's Advanced Engineering Mathematics has been updated, expanded, and now more than ever provides the solid mathematics background required throughout the engineering disciplines. Melding the author's expertise as a practitioner and his years of teaching engineering mathematics, this text stands clearly apart from the many others available. <BR><BR>Relevant, insightful examples follow nearly every concept introduced and demonstrate its practical application. This edition includes two new chapters on differential equations, another on Hilbert transforms, and many new examples, problems, and projects that help build problem-solving skills. Most importantly, the book now incorporates the use of MATLAB throughout the presentation to reinforce the concepts presented. MATLAB code is included so readers can take an analytic result, fully explore it graphically, and gain valuable experience with this industry-standard software"--Provided by publisher

"Resoundingly popular in its first edition, Dean Duffy's Advanced Engineering Mathematics has been updated, expanded, and now more than ever provides the solid mathematics background required throughout the engineering disciplines. Melding the author's expertise as a practitioner and his years of teaching engineering mathematics, this text stands clearly apart from the many others available. Relevant, insightful examples follow nearly every concept introduced and demonstrate its practical application. This edition includes two new chapters on differential equations, another on Hilbert transforms, and many new examples, problems, and projects that help build problem-solving skills. Most importantly, the book now incorporates the use of MATLAB throughout the presentation to reinforce the concepts presented. MATLAB code is included so readers can take an analytic result, fully explore it graphically, and gain valuable experience with this industry-standard software"--Provided by publisher

Includes bibliographical references and index