www.EngineeringEBooksPdf.com Advances in Applied Mathematics FOURTH EDITION Advanced Engineering Mathematics with MATLAB® Dean G Duffy www.EngineeringEBooksPdf.com Advances in Applied Mathematics Series Editor: Daniel Zwillinger Published Titles Advanced Engineering Mathematics with MATLAB, Fourth Edition Dean G Duffy CRC Standard Curves and Surfaces with Mathematica®, Third Edition David H von Seggern Dynamical Systems for Biological Modeling: An Introduction Fred Brauer and Christopher Kribs Fast Solvers for Mesh-Based Computations Maciej Paszy´nski Green’s Functions with Applications, Second Edition Dean G Duffy Introduction to Financial Mathematics Kevin J Hastings Linear and Integer Optimization: Theory and Practice, Third Edition Gerard Sierksma and Yori Zwols Markov Processes James R Kirkwood Pocket Book of Integrals and Mathematical Formulas, 5th Edition Ronald J Tallarida Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow www.EngineeringEBooksPdf.com FOURTH EDITION Advanced Engineering Mathematics with MATLAB® www.EngineeringEBooksPdf.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20161121 International Standard Book Number-13: 978-1-4987-3964-1 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been 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are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Names: Katzman, Steve (IT auditor) Title: Operational assessment of IT / Steve Katzman Description: Boca Raton, FL : CRC Press, 2016 | Series: Internal audit and IT audit ; | Includes bibliographical references and index Identifiers: LCCN 2015037136 | ISBN 9781498737685 Subjects: LCSH: Information technology Management | Information technology Auditing | Auditing, Internal Classification: LCC HD30.2 K3857 2016 | DDC 004.068 dc23 LC record available at http://lccn.loc.gov/2015037136 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.EngineeringEBooksPdf.com Dedicated to the Brigade of Midshipmen and the Corps of Cadets v www.EngineeringEBooksPdf.com www.EngineeringEBooksPdf.com Contents Dedication v Contents vii Acknowledgments xv Author xvii Introduction xix List of Definitions xxiii CLASSIC ENGINEERING MATHEMATICS 1.5 Chapter 1: First-Order Ordinary Differential Equations x 0.5 −0.5 −1 −2 −1.5 −1 −0.5 0.5 1.5 2.5 t 1.1 Classification of Differential Equations 1.2 Separation of Variables 1.3 Homogeneous Equations 16 vii www.EngineeringEBooksPdf.com viii Advanced Engineering Mathematics with MATLAB 1.4 Exact Equations 17 1.5 Linear Equations 20 1.6 Graphical Solutions 31 1.7 Numerical Methods 34 Chapter 2: Higher-Order Ordinary Differential Equations v −1 45 −2 −3 −3 −2 −1 x 2.1 Homogeneous Linear Equations with Constant Coefficients 49 2.2 Simple Harmonic Motion 57 2.3 Damped Harmonic Motion 61 2.4 Method of Undetermined Coefficients 66 2.5 Forced Harmonic Motion 71 2.6 Variation of Parameters 78 2.7 Euler-Cauchy Equation 83 2.8 Phase Diagrams 87 2.9 Numerical Methods 91 a11 a21 am1 a12 a22 am2 ··· ··· ··· a1n a2n amn Chapter 3: Linear Algebra 3.1 Fundamentals of Linear Algebra 97 97 3.2 Determinants 104 3.3 Cramer’s Rule 108 3.4 Row Echelon Form and Gaussian Elimination 111 3.5 Eigenvalues and Eigenvectors 124 3.6 Systems of Linear Differential Equations 133 3.7 Matrix Exponential 139 www.EngineeringEBooksPdf.com Table of Contents ix z (0,0,1) n C2 C3 Chapter 4: Vector Calculus (0,1,0) y 145 C1 (1,0,0) x 4.1 Review 145 4.2 Divergence and Curl 152 4.3 Line Integrals 156 4.4 The Potential Function 161 4.5 Surface Integrals 162 4.6 Green’s Lemma 169 4.7 Stokes’ Theorem 173 4.8 Divergence Theorem 179 amplitude spectrum (ft) times 10000 10000.0 1000.0 100.0 Chapter 5: Fourier Series 10.0 1.0 187 0.1 Bay bridge and tunnel 10 100 1000 10000 5.1 Fourier Series 187 5.2 Properties of Fourier Series 198 5.3 Half-Range Expansions 206 5.4 Fourier Series with Phase Angles 211 5.5 Complex Fourier Series 213 5.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations 217 5.7 Finite Fourier Series 225 4.0 3.0 y=tan( πx) Chapter 6: The Sturm-Liouville Problem x y= y 2.0 1.0 0.0 −1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 239 3.5 x 6.1 Eigenvalues and Eigenfunctions www.EngineeringEBooksPdf.com 239 966 Advanced Engineering Mathematics with MATLAB and g(x|ξ) = 2L 15 g(x|ξ) = ∞ sin(nπξ/L) sin(nπx/L) n2 π + k L2 n=1 sinh(kx< ){k cosh[k(x> − L)] − sinh[k(x> − L)]} , k sinh(kL) + k cosh(kL) and g(x|ξ) = ∞ (1 + kn2 ) sin(kn ξ) sin(kn x) , [1 + (1 + kn2 )L](kn2 + k ) n=1 where kn is the nth root of tan(kL) = −k 17 g(x|ξ) = and [a sinh(kx< ) − k cosh(kx< )] cosh[k(L − x> )] , k[a cosh(kL) − k sinh(kL)] g(x|ξ) = ∞ (a2 + kn2 ) cos[kn (ξ − L)] cos[kn (x − L)] , [(a2 + kn2 )L − a](kn2 + k ) n=1 where kn is the nth root of k tan(kL) = −a Section 15.4 ∞ t−τ nπξ g(x, t|ξ, τ ) = H(t − τ ) + H(t − τ ) cos L π n L n=1 cos nπx nπ(t − τ ) sin L L u(x, t) = ∞ sin n=1 nπ nπt e−t − cos L2 + n2 π L + L nπt sin L2 + n2 π L πx πt cos L L + sin + nπx L ∞ 4L (2m − 1)πx (2m − 1)πt sin sin π m=1 (2m − 1)2 L L u(x, t) = − ∞ t2 2L nπx − cos 2L π n=1 n2 L − cos nπt L Section 15.5 g(x, t|ξ, τ ) = L ∞ n=1 sin (2n − 1)πξ (2n − 1)πx (2n − 1)2 π (t − τ ) sin exp − 2L 2L 4L2 × H(t − τ ) www.EngineeringEBooksPdf.com Answers to the Odd-Numbered Problems ∞ u(x, t) = 2π + π n=1 ∞ 967 n nπx sin −L L n2 π e−t − exp − n2 π t L2 (2m − 1)πx (2m − 1)2 π t sin exp − 2m − L L2 m=1 ∞ u(x, t) = − t 2L nπx − cos L π n=1 n2 L g(x, y|ξ, η) = π − exp − n2 π t L2 Section 15.6 ∞ nπ nπξ exp − |y − η| sin n a a n=1 g(r, θ|ρ, θ′ ) = π ∞ nπ/β −nπ/β nπθ′ r< r> sin n β n=1 sin sin nπx a nπθ β g(r, z|ρ, ζ) = ∞ ∞ J0 (km ρ/a)J0 (km r/a) nπζ sin 2 2 2 2 πa L n=1 m=1 πa LJ1 (km )(km /a + n π /L ) L sin nπz L Section 16.2 (a) S = {HH, HT, T H, T T } (b) S = {ab, ac, ba, bc, ca, cb} (c) S = {aa, ab, ac, ba, bb, bc, ca, cb, cc} (d) S = {bbb, bbg, bgb, bgg, ggb, ggg, gbb, gbg} (e) S = {bbb, bbg, bgb, bgg, ggb, ggg, gbb, gbg} 1/3 1/3 2/13 11 1/2 13 1/2 15 9/16 1/720, 1/120 Section 16.3 FX (x) = 0, − p, 1, x < 0, ≤ x < 1, ≤ x www.EngineeringEBooksPdf.com 27 968 Advanced Engineering Mathematics with MATLAB Section 16.4 FX (x) = 0, (1 + x)2 /2, FX (x) = − (x − 1) /2, 1, x ≤ 0, < x 0, − e−λx , Section 16.5 E(X) = 12 , and Var(X) = φX (ω) = peiω + q n x < −1, −1 ≤ x < 0, ≤ x < 1, ≤ x k = 3/4, E(X) = 1, and Var(X) = , µX = np, Var(X) = npq φX (ω) = p/(1 − qeωi ), µX = q/p, Var(X) = q/p2 Section 16.6 (a) 1/16, (b) 1/4, (c) 15/16, (d) 1/16 P (T < 150) = 13 , and P (X = 3) = 0.1646 P (X > 0) = 0.01, and P (X > 1) = × 10−5 Section 16.7 pXY [xi , yj ] = xi yj 20 5 − x i − yj , where xi = 0, 1, 2, 3, 4, 5, yj = 0, 1, 2, 3, 4, and ≤ xi + yj ≤ Section 17.1 µX (t) = 0, and σX (t) = cos(ωt) For t1 = t2 , RX (t1 , t2 ) = p; for t1 = t2 , RX (t1 , t2 ) = p2 For t1 = t2 , CX (t1 , t2 ) = p(1−p); for t1 = t2 , CX (t1 , t2 ) = Section 17.4 Pn = 2/3 + (1/3)(1/4)n 2/3 − (2/3)(1/4)n 1/3 − (1/3)(1/4)n 1/3 + (2/3)(1/4)n P∞ = 2/3 2/3 1/3 1/3 Section 18.3 E(X) = 0, E(X) = 0, Var(X) = E(X ) = (b3 − a3 )/3 Var(X) = E(X ) = (b2 − a2 )/2 Section 18.5 X(t) = et/2 B(t) + X0 et/(RC) Q(t) − Q(0) = X(t) = B (t) + tB(t) + X0 R t η/(RC) e V (η) dη + R X(t) = tB (t) − t2 /2 + X0 t η/(RC) e α(η) dη www.EngineeringEBooksPdf.com Answers to the Odd-Numbered Problems X(t) = X(0)e4t + − e−4t + t 4(t−η) e 969 dB(η) 11 X(t) = X0 et + et − + 12 et [B (t) − t] 13 X(t) = et/2 X0 − 2t − + 5et/2 − et cos[B(t)] 17 t X(t) = X0 exp sin(2t) + sin(t) − 14 t + sin(η) dB(η) 19 t X(t) = X0 exp (t + 1) ln(t + 1) − t − 16 t3 + www.EngineeringEBooksPdf.com η dB(η) www.EngineeringEBooksPdf.com Index abscissa of convergence, 560 absolute value of a complex number, 442 Adams-Bashforth method, 40 addition of a complex numbers, 441 of matrices, 99 of vectors, 145 age of the earth, 553–554 aliasing, 231–233 amplitude of a complex number, 442 spectrum, 511 analytic complex function, 449 derivative of, 450 analytic signals, 718–720 Archimedes’ principle, 184–185 argument of a complex number, 442 autonomous ordinary differential equation, 4, 48 auxiliary equation, 50 back substitution, 102, 113 band-pass functions, 717 basis function, 795 Bayes’ rule, 812 Bernoulli equation, 28–29 Bessel equation of order n, 271–275 function of the first kind, 273 expansion in, 277–285 function of the second kind, 273 function, modified, 275 recurrence formulas, 276–277 Bessel, Friedrich Wilhelm, 272 Biot number, 352 boundary condition Cauchy, 300 Dirichlet, 339 Neumann, 339 Robin, 340 boundary-value problems, 46 branches of a complex function, 449 principal, 443 Bromwich contour, 613 Bromwich integral, 613 Bromwich, Thomas John I’Anson, 614 Buffon’s needle problem, 846–847 carrier frequency, 524 Cauchy boundary condition, 300 data, 300 integral formula, 464–466 principal value, 485–488 problem, 300 residue theorem, 473–476 Cauchy, Augustin-Louis, 451 971 www.EngineeringEBooksPdf.com 972 Advanced Engineering Mathematics with MATLAB Cauchy-Goursat theorem, 460 Cauchy-Riemann eqns, 451 centered finite differences, 326 central limit theorem, 827 Chapman-Kolmogorov equation, 868–869 characteristic polynomial, 124 equation, 50 value, 124, 240 vector, 124 characteristic function, 240, 830–831 characteristics, 318 chemical reaction, 12–13, 882–883 circular frequency, 58 circulation, 159 closed contour integral, 157, 459 surface integral, 163 coefficient matrix, 112 cofactor, 105 column of a matrix, 98 column vector, 98 combinations, 811 complementary error function, 565 complementary solution of an ordinary differential equation, 66 complex-valued function, 448–450 conjugate, 441 number, 441 plane, 442 variable, 441 complex matrix, 98 components of a vector, 145 compound interest, 9, 694–695 conformable for addition of matrices, 99 for multiplication of matrices, 99 conformal mapping, 491–507 conservative field, 159 consistency in finite differencing for the heat equation, 378 for the wave equation, 329 consistent system of linear eqns, 111 contour integrals, 456–460 convergence of a Fourier integral, 512 of finite difference solution for heat equation, 379 for wave equation, 330 of Fourier series, 189 convolution theorem for Fourier transforms, 544–547 for Hilbert transforms, 715–717 for Laplace transforms, 588–591 for z-transforms, 678 Coriolis force, 147 Cramer’s rule, 108 Crank-Nicholson method, 382 critical points, 33, 88 stable, 33, 89 stable node, 90 unstable, 33, 89 cross product, 146 curl, 154 curve, simply closed, 460 space, 146 cutoff frequency, 754 d’Alembert’s formula, 320 d’Alembert’s solution, 318–324 d’Alembert, Jean Le Rond, 319 damped harmonic motion, 61, 899–900 damping constant, 61 de Moivre’s theorem, 443 deformation principle, 462 degenerate eigenvalue problem, 247 del operator, 148 delay differential equation, 608–609 delta function, 512–513, 568–570 design of film projectors, 585–587 design of wind vane, 64–65 determinant, 104–108 diagonal, principal, 98 difference eqns, 667 differential eqns nth order, 45–95 first-order, 1–43 nonlinear, order, ordinary, 1–97 partial, 1, 297–336, 337–384, 385–440 stochastic, 929–932 type, differentiation of a Fourier series, 199 diffusivity, 338 www.EngineeringEBooksPdf.com Index 973 dimension of a vector space, 125 direction fields, 31 Dirichlet conditions, 189 Dirichlet problem, 339 Dirichlet, Peter Gustav Lejeune, 191 dispersion, 306 divergence of a vector, 153 theorem, 179–185 division of complex numbers, 441 dot product, 146 double Fourier series, 427 dual Fourier-Bessel series, 404 dual integral eqns, 401 Duhamel’s theorem for ordinary differential equation, 741 for the heat equation, 651–659 eigenfunctions, 239–256 expansion in, 251 orthogonality of, 249 eigenvalue(s) of a matrix, 124 of a Sturm-Liouville problem, 239–247 eigenvalue problem, 124–129 for ordinary differential eqns, 239–247 singular, 240 eigenvectors, 124–129 orthogonality of, 132 electrical circuits, 24, 76, 604–609 electrostatic potential, 392 element of a matrix, 98 elementary row operations, 111 elliptic partial differential equation, 385 entire complex function, 449 equilibrium points, 33, 88 equilibrium systems of linear eqns, 111 error function, 565 essential singularity, 469 Euler’s formula, 442 Euler’s method, 34–36 Euler-Cauchy equation, 83–86 evaluation of partial sums using z-transform, 688 exact ordinary differential equation, 17 existence of ordinary differential eqns nth-order, 46 first-order, explicit numerical methods for the heat equation, 377 for the wave equation, 326 exponential order, 560 fast Fourier transform (FFT), 231 filter, 234 final-value theorem for Laplace transforms, 575 for z-transform, 676 finite difference approximation to derivatives, 326 finite element, 795 finite Fourier series, 225–234 first-order ordinary differential eqns, 1–43 linear, 20–31 first-passage problem, 901–903 flux lines, 150 folding frequency, 233 forced harmonic motion, 71–76 Fourier coefficients, 188 cosine series, 194 cosine transform, 555 Joseph, 190 number, 347 series for a multivalued function, 217 series for an even function, 194 series for an odd function, 194 series in amplitude/phase form, 211–213 series on [−L, L], 187–198 sine series, 194 sine transform, 555 Fourier coefficients, 253 Fourier cosine series, 194 Fourier transform, 509–549, 736–740 basic properties of, 520–530 convolution, 544–547 inverse of, 510, 532–542 method of solving the heat eqn, 551–556 of a constant, 518 of a derivative, 524 of a multivariable function, 514 of a sign function, 519 of a step function, 520 Fourier-Bessel coefficients, 279 expansions, 277 Fourier-Legendre coefficients, 264 expansion, 263 Fredholm integral eqn, 121 www.EngineeringEBooksPdf.com 974 Advanced Engineering Mathematics with MATLAB free umderdamped motion, 58 frequency convolution, 546 frequency modulation, 526 frequency response, 735 frequency spectrum, 511 for a damped harmonic oscillator, 736–737 for low-frequency filter, 738–739 function even extension of, 206 generalized, 570 multiplied complex, 448 odd extension of, 206 single-valued complex, 448 vector-valued, 148 fundamental of a Fourier series, 188 Galerkin method, 795–801 gambler’s ruin problem, 858 Gauss’s divergence theorem, 179–185 Gauss, Carl Friedrich, 180 Gauss-Seidel method, 429 Gaussian elimination, 114 general solution to an ordinary differential equation, generalized Fourier series, 252 generalized functions, 570 generating function for Legendre polynomials, 260 Gibbs phenomenon, 202–205, 267 gradient, 148 graphical stability analysis, 33 Green’s function, 725–742 for a damped harmonic oscillator, 737 for low-frequency filter, 738 Green’s lemma, 169–172 grid point, 325 groundwater flow, 388–392 half-range expansions, 206–209 Hankel transform, 399 harmonic functions, 386, 455 complex, 455 harmonics of a Fourier series, 188 heat conduction in a rotating satellite, 220–224 within a metallic sphere, 409–414 heat dissipation in disc brakes, 640–642 heat equation, 337–383, 551–556, 637–662 for a semi-infinite bar, 551–553 for an infinite cylinder, 357–360 nonhomogeneous, 339 one-dimensional, 340–343 within a solid sphere, 355–357 Heaviside expansion theorem, 581–587 step function, 563–565 Heaviside, Oliver, 564 Hilbert pair, 704 Hilbert transform, 703–723 and convolution, 715–716 and derivatives, 714–715 and shifting, 714 and time scaling, 714 discrete, 711–712 linearity of, 713 product theorem, 716–717 Hilbert, David, 705 holomorphic complex function, 449 homogeneous ordinary differential eqns, 16–17, 45 solution to ordinary differential eqn, 66 system of linear eqns, 101 hydraulic potential, 338 hydrostatic equation, hyperbolic partial differential equation, 297 ideal Hilbert transformer, 703 ideal sampler, 668 imaginary part of a complex number, 441 importance sampling, 833 impulse function see (Dirac) delta function impulse response, 732 inconsistent system of linear eqns, 111 indicial admittance for ordinary differential eqns, 733–734 of heat equation, 654 inertia supercharging, 208 initial -value problem, 45, 597–610 conditions, 300 initial-boundary-value problem, 339 initial-value theorem for Laplace transforms, 574 for z-transforms, 676 inner product, 99 integral curves, 87 www.EngineeringEBooksPdf.com Index 975 integral equation, 731 of convolution type, 592–594 integrals complex contour, 456–460 Fourier type, evaluation of, 536–537 line, 156–159 real, evaluation of, 477–482 integrating factor, 19 integration of a Fourier series, 200–201 interest rate, 9, 694 inverse discrete Fourier transform, 226 Fourier transform, 510, 532–542 Hilbert transform, 704 Laplace transform, 581–588, 613–617 z-transform, 681–689 inversion formula for the Fourier transform, 510 for the Hilbert transform, 704 for the Laplace transform, 613–617 for the z-transform, 681–689 inversion of Fourier transform by contour integration, 533–542 by direct integration, 532 by partial fraction, 532–533 inversion of Laplace transform by contour integration, 614–617 by convolution, 588 by partial fractions, 581–583 in amplitude/phase form, 583–587 inversion of z-transform by contour integration, 685–689 by partial fractions, 683–685 by power series, 681–683 by recursion, 682–683 irrotational, 154 isoclines, 31 isolated singularities, 452 iterative methods Gauss-Seidel, 429 successive over-relaxation, 432 iterative solution of the radiative transfer equation, 267–269 Itˆo process, 896 Itˆo’s integral, 916–921 Itˆo’s lemma, 920–928 Itˆo, Kiyhosi, 923 joint transform method, 752 Jordan curve, 460 Jordan’s lemma, 533 Kirchhoff’s law, 24 Klein-Gordon equation, 307 Kramers-Kronig relationship, 721–223 Lagrange’s trigonometric identities, 445 Laguerre polynomial, 596 Laplace integral, 559 Laplace transform, 559–619 basic properties of, 571–577 convolution for, 588–591 definition of, 559 derivative of, 573 in solving delay differential equation, 608–609 heat equation, 637–644 integral eqns, 592–594 Laplace equation, 662–664 wave equation, 619–629 integration of, 574 inverse of, 581–587, 613–617 of derivatives, 562 of periodic functions, 579–581 of the delta function, 568–570 of the step function, 563–565 Schouten-van der Pol theorem for, 619 solving of ordinary differential eqns, 597–610 Laplace’s eqn, 385–433, 550–551, 662–664 finite element solution of, 433 in cylindrical coordinates, 386 in spherical coordinates, 387 numerical solution of, 428–433 solution by Laplace transforms, 662–664 solution by separation of variables, 388–415 solution on the half-plane, 549–551 Laplace’s expansion in cofactors, 105 Laplace, Pierre-Simon, 387 Laplacian, 153 Laurent expansion, 469 Lax-Wendroff scheme, 334 Legendre polynomial, 259 expansion in, 263 generating function for, 260–261 orthogonality of, 263 recurrence formulas, 261 Legendre’s differential equation, 257 Legendre, Adrien-Marie, 257 www.EngineeringEBooksPdf.com 976 Advanced Engineering Mathematics with MATLAB length of a vector, 145 line integral, 156–159, 456–460 line spectrum, 215 linear dependence of eigenvectors, 124 of functions, 53 linear transformation, 103 linearity of Fourier transform, 520 of Hilbert transform, 713 of Laplace transform, 561 of z-transform, 674 lines of force, 150 Liouville, Joseph, 241 logistic equation, 12 low-frequency filter, 738–739 low-pass filter, 900–902 LU decomposition, 122 magnitude of a vector, 145 Markov chain state, 867 state transition, 867 time homogeneous, 868 matrices addition of, 99 equal, 98 multiplication, 99 matrix, 97 algebra, 97 amplification, 129 augmented, 112 banded, 102 coefficient, 112 complex, 98 diagonalization, 138 exponential, 139 identity, 98 inverse, 100 invertible, 100 method of stability of a numerical scheme, 128 nonsingular, 100 null, 98 orthogonal, 123 real, 98 rectangular, 98 square, 98 symmetric, 98 tridiagonal, 102 unit, 98 upper triangular, 102 zero, 98 matrix exponential, 139 maximum principle, 386 Maxwell’s field eqns, 156 mean, 828–830 mechanical filter, 587 meromorphic function, 452 method of partial fractions for Fourier transform, 532–533 for Laplace transform, 581–587 for z-transform, 683–685 method of undetermined coefficients, 67-70 minor, 105 mixed boundary-value problems, 400–405 modified Bessel function, first kind, 275 second kind, 275 Euler method, 34–36 modified Euler method, 34 modulation, 524–527 modulus of a complex number, 442 Monte Carlo integration, 833 multiplication of complex numbers, 441 of matrices, 99 multivalued complex function, 448 nabla operator, 148 natural vibrations, 306 Neumann problem, 339 Neumann’s Bessel function of order n, 274 Newton’s law of cooling, 351 nonanticipating process, 917 nondivergent, 153 nonhomogeneous heat equation, 339 ordinary differential equation, 45 system of linear eqns, 101 norm of a vector, 98, 145 normal differential equation, 45 normal mode, 306 normal to a surface, 148 not simply connected, 460 numerical solution of heat equation, 377–383 of Laplace’s equation, 428–433 of stochastic differential eqn, 936–943 of the wave equation, 326–334 www.EngineeringEBooksPdf.com Index 977 Nyquist frequency, 233 Nyquist sampling criteria, 231 one-sided finite difference, 326 order of a matrix, 98 of a pole, 470 orthogonal matrix, 123 orthogonality, 249 of eigenfunctions, 249–251 of eigenvectors, 132 orthonormal eigenfunction, 251 overdamped ordinary differential eqn, 62 overdetermined system of linear eqns, 116 parabolic partial differential eqn, 338 Parseval’s equality, 201–202 Parseval’s identity for Fourier series, 201 for Fourier transform, 527–529 for z-transform, 688 partial fraction expansion for Fourier transform, 532 for Laplace transform, 581–587 for z-transform, 683–685 particular solution to ordinary differential equation, 3, 66 path in complex integrals, 457 in line integrals, 157 path independence in complex integrals, 462 in line integrals, 159 permutation, 811 phase, 442 angle in Fourier series, 211–213 diagram, 87 line, 33 path, 87 spectrum, 511 phase of the complex number, 442 phasor amplitude, 721 pivot, 112 pivotal row, 112 Poisson process, 886–891 arrival time, 889 Poisson’s equation, 425–427 integral formula for a circular disk, 414–415 for upper half-plane, 551 summation formula, 529–532 Poisson’s summation formula, 529 Poisson, Sim´eon-Denis, 426 polar form of a complex number, 442 pole of order n, 470 population growth and decay, 11, 874–883 position vector, 145 positive oriented curve, 463 potential flow theory, 155 potential function, 161–162 power content, 201 power spectrum, 528, 864–867 principal branch, 443 principal diagonal, 98 principle of linear superposition, 50, 303 probability Bernoulli distribution, 834 Bernoulli trials, 820 binomial distribution, 836 characteristic function, 830 combinations, 811 conditional, 812 continuous joint distribution, 844 correlation, 850 covariance, 848 cumulative distribution, 824 distribution function, 824 event, 806 elementary, 806 simple, 806 expectation, 828 experiment, 805 exponential distribution, 839 Gaussian distribution, 840 geometric distribution, 835 independent events, 814 joint probability mass function, 842 law of total probability, 813 marginal probability functions, 842 mean, 828 normal distribution, 840 permutation, 811 Poisson distribution, 837 probability integral, 841 probability mass function, 818 random variable, 817–818 sample point, 805 sample space, 805 standard normal distribution, 841 uniform distribution, 838 www.EngineeringEBooksPdf.com 978 Advanced Engineering Mathematics with MATLAB probability (continued) variance, 828 QR decomposition, 123 quadrature phase shifting, 703 quieting snow tires, 195–198 radiation condition, 300, 351 radius of convergence, 467 random differential equation, 897–898 random process, 855–891 autocorrelation function, 860 Bernoulli process, 856 Brownian motion, 905–913 chemical kinetics, 879 counting process, 857 mean, 858 power spectrum, 864 realization, 855 sample function, 855 sample path, 855 state, 855 state space, 855 variance, 858 wide-sense stationary processes, 862 Wiener process, 912–913 random variable, 817 discrete, 818 domain, 817 range, 817 rank of a matrix, 114 real definite integrals evaluation of, 477–482 real matrix, 98 real part of a complex number, 441 rectangular matrix, 98 recurrence relation for Bessel functions, 276–277 for Legendre polynomial, 261–263 in finite differencing, 92 reduction in order, 47 regular complex function, 449 regular Sturm-Liouville problem, 240 relaxation methods, 429–433 removable singularity, 470 residue, 469 residue theorem, 472–476 resonance, 75, 219, 600 rest points, 33 Riemann, Georg Friedrich Bernhard, 452 Robin problem, 340 Rodrigues’ formula, 260 root locus method, 737 roots of a complex number, 445–447 row echelon form, 113 row vector, 98 rows of a matrix, 98 Runge, Carl, 38 Runge-Kutta method, 37–40, 93–97 scalar, 145 Schouten-Van der Pol theorem, 619 Schwarz-Christoffel transformation, 496–507 Schwarz’ integral formula, 551 second shifting theorem, 565 secular term, 219 separation of variables for heat equation, 340–366 for Laplace’s equation, 388–415 for ordinary differential eqns, 4-14 for Poisson’s equation, 425–427 for wave equation, 300–314 set, 804 complement, 804 disjoint, 804 element, 804 empty, 804 intersection, 804 null, 804 subset, 804 union, 804 universal, 804 shifting in the ω variable, 525 in the s variable, 571 in the t variable, 521, 572 sifting property, 513 simple eigenvalue, 242 pole, 470 simple harmonic motion, 58, 600 simple harmonic oscillator, 57–61 simply closed curve, 460 sinc function, 511 single side-band signal, 720 single-valued complex function, 448 singular solutions to ordinary differential eqns, Sturm-Liouville problem, 240 www.EngineeringEBooksPdf.com Index 979 singular Sturm-Liouville problem, 240 singular value decomposition, 132 singularity essential, 469 isolated, 469 pole of order n, 470 removable, 470 slope field, 31 solenoidal, 153 solution curve, 31 solution of ordinary differential eqns by Fourier series, 217–224 by Fourier transform, 547–549, 735–740 space curve, 146 spectral radius, 124 spectrum of a matrix, 124 square matrix, 98 stability of numerical methods by Fourier method for heat eqn, 379 by Fourier method for wave eqn, 329 by matrix method for wave equation, 128 steady-state heat equation, 10, 347 steady-state output, 33 steady-state solution to ordinary differential eqns, 73 steady-state transfer function, 735 step function, 563–565 step response, 733 stochastic calculus, 895–941 Brownian motion, 906–913 damped harmonic motion, 899–900 derivative, 895 differential eqns, 928–932 first-passage problem, 901–903 integrating factor, 930 Itˆ o process, 896 Itˆ o’s integral, 916–921 Itˆ o’s lemma, 920–928 low-pass filter, 900–902 nonlinear oscillator, 941 numerical solution, 936–939 Euler-Marugama method, 937 Milstein method, 938 product rule, 926, 929 random differential eqns, 897–898 RL electrical circuit with noise, 939 wave motion due to random forcing, 904–906 Wiener process, 913 stochastic process, 855 Stokes’ theorem, 173–178 Stokes, Sir George Gabriel, 174 streamlines, 150 Sturm, Charles-Fran¸cois, 240 Sturm-Liouville equation, 240 problem, 239–247 subtraction of complex numbers, 441 of matrices, 99 of vectors, 145 successive over-relaxation, 429 superposition integral, 727 for ordinary differential eqns, 741 of heat equation, 651–659 superposition principle, 303 surface conductance, 351 surface integral, 162–166 system of linear differential eqns, 133–137 homogeneous eqns, 101 nonhomogeneous eqns, 101 tangent vector, 146 Taylor expansion, 467 telegraph equation, 309, 620–629 telegraph signal, 856, 863 terminal velocity, 9, 27 thermal conductivity, 338 threadline equation, 299–300 time shifting, 521, 571 trajectories, 87 transfer function, 732 transform Fourier, 509–549, 736–740 Hilbert, 703–723 Laplace, 559–619 z-, 667–702 transient solution to ordinary differential eqns, 73 transmission line, 620–629 transmission probability matrix, 869 transpose of a matrix, 101 tridiagonal matrix, solution of, 101–102 underdamped, 62 underdetermined system of linear eqns, 113 www.EngineeringEBooksPdf.com 980 Advanced Engineering Mathematics with MATLAB uniformitarism, 554 uniqueness of ordinary differential eqns nth-order, 46 first-order, unit normal, 149 step function, 563–565 vector, 145 Vandermonde’s determinant, 108 variance, 828–830 variation of parameters, 78–83 vector, 98, 145 vector element of area, 165 vector space, 125 Venn diagram, 804 vibrating string, 297–299 vibrating threadline, 299–300 vibration of floating body, 60 Volterra equation of the second kind, 592 volume integral, 179–185 wave equation, 297–334, 619–635 damped, 308–311 for a circular membrane, 312–314 for an infinite domain, 318–324 one-dimensional, 299 wave motion due to random forcing, 904–906 weight function, 249 Wiener process, 860, 913 Wiener, Norbert, 911 Wronskian, 54 z-transform, 667–702 basic properties of, 674–680 convolution for, 678 final-value theorem for, 676–677 for solving difference eqns, 691–697 initial-value theorem for, 676 inverse of, 681–689 linearity of, 674 multiplication by n, 677 of a sequence multiplied by an exponential sequence, 674 of a shifted sequence, 674–676 of periodic sequences, 677–678 their use in determining stability, 697–702 zero vector, 145 www.EngineeringEBooksPdf.com