applied mathematics for engineers and physicists third edition pdf

Applied Mathematics for Engineers and Physicists Third Edition Louis A Pipes and Lawrence R Harvill DOVER PUBLICATIONS, INC MINEOLA, NEW YORK www.TechnicalBooksPDF.com Copyright Copyright © 1946, 1958, 1970 by Louis A Pipes and Lawrence R Harvill Copyright © 1998 by Johanna I Pipes and Lawrence R Harvill Preface to the Dover Edition copyright © 2014 by Lawrence R Harvill All rights reserved Bibliographical Note This Dover edition, first published in 2014, is an unabridged republication of the work originally published by the McGraw-Hill Book Company, New York, in 1970 Lawrence R Harvill has prepared a new Preface specially for this Dover edition Library of Congress Cataloging-in-Publication Data Pipes, Louis Albert, 1910– author Applied mathematics for engineers and physicists / Louis A Pipes and Lawrence R Harvill — Third edition pages cm Reprint of: 3rd ed — New York : McGraw-Hill, [1970] eISBN-13: 978-0-486-79499-0 Mathematical physics Mechanics, Applied I Harvill, Lawrence R., 1935– author II Title QA401.P5 2014 510.24’53—dc23 2013043789 Manufactured in the United States by Courier Corporation 77951301 2014 www.doverpublications.com www.TechnicalBooksPDF.com PREFACE TO THE DOVER EDITION This book represents the reprinting of the third edition of Applied Mathematics for Engineers and Physicists, originally published in 1970 as part of the International Series in Pure and Applied Mathematics The first edition was published in 1946 and the second in 1958, both by the primary author, Louis A Pipes It is one of three major texts for engineers and physicists with the other two being Advanced Engineering Mathematics by Erwin Kreyszig and Mathematics of Physics and Modern Engineering by Ivar Stephen Sokolnikoff and Raymond M Redheffer Louis or “Louie” as he was called by his friends and colleagues was a quintessential professor and avid student of the applications of mathematics to a wide variety of problems In addition to his scholarly research he was a consultant for the Aerospace Corporation in Redondo Beach, CA and the Naval Weapons Center in Ridgecrest, CA In the classroom Louie was an amazingly effective teacher with an exceptional memory of all the primary literature in the area he was presenting Hardly a lecture would go by without a reference to a story about a humorous event or a quote by the author whom he was presenting His lectures were so clear and logical, that it was easy to be lulled into a false sense of security, feeling that one fully understood the subject After class, one quickly realized that taking good notes was the only way to avoid extra hours struggling to work out the homework Louie also had a great sense of humor Sometimes during a lecture when a student asked a question that was slightly off, Louie would stare at the student and his face would assume a questioning scowl; he would then reach into his coat pocket and pull out a cap pistol and fire at the stunned student The scowl would dissolve into a smile and after the gales of laughter subsided; he would go about a careful explanation of the point the student had missed After completing my PhD under Louie’s guidance I accepted a faculty position in engineering at the University of Redlands in 1964 Shortly thereafter Louie asked me to coauthor this third edition, which initiated a long period of close collaboration and enjoyable tennis matches until the work was published in 1970 It was later translated into a four volume Japanese edition Throughout my 34-year career of teaching at the University of Redlands I always tried to model my teaching after Louie’s exceptional style and obvious great enthusiasm for the subject at hand L HARVILL, PHD, PE Redlands, CA, 2014 www.TechnicalBooksPDF.com PREFACE The first edition of this text was published 23 years ago and the second edition 11 years ago It was stated in the Preface to the second edition that the tremendous development of high-speed computing devices was a major factor guiding the changes and revisions presented in that edition The developments which have come about in digital computers since the publication of the second edition have been all the more spectacular Computation speeds and memory size have increased by two orders of magnitude during this period, while machine costs have dropped by an order of magnitude A single modern center, such as the Western Data Processing Center at the University of California at Los Angeles, has as much computing capacity today as all the combined installations in the United States a decade ago There is also no apparent reason why one should not expect these trends to continue over the next decade Another factor to be considered is the recent development of time-sharing computer systems which, through large numbers of remote terminals, provide engineers and scientists with direct and immediate access to a computer for problem solving It may not be too long until the remote terminal has replaced the slide rule as a readily available and sophisticated computing tool As a result the analyst will be spending more time on developing realistic mathematical models for a physical problem and less time on the computational details Also, the magnitude of the computations are of lesser concern because of the great speed of digital machines However, computers have not eliminated the problem of choosing between accuracy and speed; they have only shifted the breakeven point to a higher level We would like to suggest two particular areas of concern which we feel should be reviewed from time to time by every analyst The first is to maintain an awareness of the limitations of any mathematical model resulting from the various approximations imposed during the modeling process This is very important in order to avoid predicting the behavior of a system by a solution obtained from a model based on postulates which are invalid in the region of interest Our second concern is that even though computers have given us the ability to study complex models, we should not stop seeking simpler representations, as it is an easy matter to overcomplicate a problem Perhaps the greatest factor creating the need for the extensive changes included in this third edition has been the widespread changes in engineering curricula which will continue for several years to come Coupled with this are improvements in teaching mathematics at the high school level, which have enabled the shifting of more advanced material into the lower-division mathematics sequences in many colleges These two situations have increased the mathematics requirements for engineering students and raised the level of mathematical rigor The authors feel that both of these situations are desirable, but it was decided not to trade off the valuable physical applications for increased rigor in this edition It is felt that most instructors teaching an advanced course based on this text can easily add any desired amount of rigor during the lectures but that it is usually more difficult to add a wide range of physical applications during the lecture period Those readers who are familiar with the previous two editions of this book will rapidly become aware of the extensive changes which have been made in this third edition The arrangement of the chapters has been changed greatly, with those on series, special functions, vectors and tensors, transcendental equations, and partial differentiation being moved to appendixes This change is an attempt to make the material more flexible for the variety of courses in which this book could be employed as a text Since the material in the body of the text is designed for use in a one-year course, www.TechnicalBooksPDF.com it is hoped that each instructor will feel free to add from, or exchange with, the material in the appendixes to meet the level and individual requirements of his course Other changes are the combining of the two previous tables of Laplace transforms into a single one i n Appendix A, to which a few additional transform pairs have been added The p-multiplied Laplace-transform notation has been dropped in favor of the more common s notation Also additional problems have been added to each chapter, with answers and hints to selected problems supplied in Appendix G Fourier, Hankel, and Mellin transforms have been added to the chapter on operational methods, and a new chapter on statistics and probability has been included As previously mentioned, this text is designed for use in a one-year course, and the chapters have been written to make them as independent as possible to give each instructor freedom in designing his own course Chapters to are primarily concerned with the analysis of lumped parameter systems and could comprise the material for the first semester’s course with or without some combinations of material contained in the appendixes Chapters to 13 deal with distributed parameter systems, while Chapters 14 to 16 cover various important areas of applied mathematics As such, these eight chapters could comprise the material for the second-semester’s course The authors would like to extend their appreciation to colleagues and students for their helpful assistance and insights in the preparation of this material Thanks are also due to the editors of McGraw-Hill for their patience and assistance, and to the reviewers who contributed several helpful suggestions for improvement Special thanks are due to Professor S Takeda, Hosei University, Tokyo, for his detailed errata for the second edition which have been included in this present edition Finally the greatest acknowledgment is due to our wives, Johanna and Doris, for their continued encouragement and understanding LOUIS A PIPES LAWRENCE R HARVILL www.TechnicalBooksPDF.com CONTENTS Chapter THE THEORY OF COMPLEX VARIABLES Introduction Functions of a Complex Variable The Derivative and the Cauchy-Riemann Differential Equations Line Integrals of Complex Functions Cauchy’s Integral Theorem Cauchy’s Integral Formula Taylor’s Series Laurent’s Series Residues: Cauchy’s Residue Theorem 10 Singular Points of an Analytic Function 11 The Point at Infinity 12 Evaluation of Residues 13 Liouville’s Theorem 14 Evaluation of Definite Integrals 15 Jordan’s Lemma 16 Bromwich Contour Integrals 17 Integrals Involving Multiple-valued Functions (Branch Points) 18 Further Examples of Contour Integrals Around Branch Points 19 The Use of z and in the Theory of Complex Variables Problems References Chapter LINEAR DIFFERENTIAL EQUATIONS Introduction The Reduced Equation; the Complementary Function Properties of the Operator Ln(D) The Method of Partial Fractions Linear Dependence: Wronskian The Method of Undetermined Coefficients The Use of Complex Numbers to Find the Particular Integral Linear Second-order Differential Equations with variable Coefficients The Method of Frobenius 10 Variation of Parameters 11 The Sturm-Liouville Differential Equation Problems References Chapter www.TechnicalBooksPDF.com LINEAR ALGEBRAIC EQUATIONS, DETERMINANTS, AND MATRICES Introduction Simple Determinants Fundamental Definitions Laplace Expansion Fundamental Properties of Determinants The Evaluation of Numerical Determinants Definition of a Matrix Special Matrices Equality of Matrices; Addition and Subtraction 10 Multiplication of Matrices 11 Matrix Division, the Inverse Matrix 12 The Reversal Law in Transposed and Reciprocated Products 13 Properties of Diagonal and Unit Matrices 14 Matrices Partitioned into Submatrices 15 Matrices of Special Types 16 The Solution of Linear Algebraic Equations 17 The Special Case of n Equations and n Unknowns 18 Systems of Homogeneous Linear Equations 19 The Characteristic Matrix and the Characteristic Equation of a Matrix 20 Eigenvalues and the Reduction of a Matrix to Diagonal Form 21 The Trace of a Matrix 22 The Cayley-Hamilton Theorem 23 The Inversion of Large Matrices 24 Sylvester’s Theorem 25 Power Series of Matrices; Functions of Matrices 26 Alternate Method of Evaluating Functions of Matrices 27 Differentiation and Integration of Matrices 28 Association of Matrices with Linear Differential Equations 29 Method of Peano-Baker 30 Adjoint Method 31 Existence and Uniqueness of Solutions of Matrix Differential Equations 32 Linear Equations with Periodic Coefficients 33 Matrix Solution of the Hill-Meissner Equation 34 The Use of Matrices to Determine the Roots of Algebraic Equations Problems References Chapter LAPLACE TRANSFORMS Introduction The Fourier-Mellin Theorem The Fundamental Rules www.TechnicalBooksPDF.com Calculation of Direct Transforms Calculation of Inverse Transforms The Modified Integral Impulsive Functions Heaviside’s Rules The Transforms of Periodic Functions 10 The Simple Direct Laplace-transform, or Operational, Method of Solving Linear Differential Equations with Constant Coefficients 11 Systems of Linear Differential Equations with Constant Coefficients Problems References Chapter OSCILLATIONS OF LINEAR LUMPED ELECTRICAL CIRCUITS Introduction Electrical-circuit Principles Energy Considerations Analysis of General Series Circuit Discharge and Charge of a Capacitor Circuit with Mutual Inductance Circuits Coupled by a Capacitor The Effect of Finite Potential Pulses Analysis of the General Network 10 The Steady-state Solution 11 Four-terminal Networks in the Alternating-current Steady State 12 The Transmission Line as a Four-terminal Network Problems References Chapter OSCILLATIONS OF LINEAR MECHANICAL SYSTEMS Introduction Oscillating Systems with One Degree of Freedom Two Degrees of Freedom Lagrange’s Equations Proof of Lagrange’s Equations Small Oscillations of Conservative Systems Solution of the Frequency Equation and Calculation of the Normal Modes by the Use of Matrices Numerical Example: the Triple Pendulum Nonconservative Systems: Vibrations with Viscous Damping 10 A Matrix Iterative Method for the Analysis of Nonconservative Systems 11 Forced Oscillations of a Nonconservative System Problems References www.TechnicalBooksPDF.com Chapter THE CALCULUS OF FINITE DIFFERENCES AND LINEAR DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS Introduction The Fundamental Operators of the Calculus of Finite Differences The Algebra of Operators Fundamental Equations Satisfied by the Operators Difference Tables The Gregory-Newton Interpolation Formula The Derivative of a Tabulated Function The Integral of a Tabulated Function A Summation Formula 10 Difference Equation with Constant Coefficients 11 Oscillations of a Chain of Particles Connected by Strings 12 An Electrical Line with Discontinuous Leaks 13 Filter Circuits 14 Four-terminal-network Connection with Matrix Algebra 15 Natural Frequencies of the Longitudinal Motions of Trains Problems References Chapter TRANSFER FUNCTIONS AND IMPULSE RESPONSES Introduction Transfer Functions of Linear Systems Solutions to Problems Using Transfer Functions Combining Transfer Functions of Several Systems Matrix Method for Evaluating Over-all Transfer Functions When Loading Occurs Method Impulse Responses and Transfer Functions Feedback Control in Linear Systems Stability of Linear Systems Problems References Chapter LAPLACE’S EQUATION Introduction Laplace’s Equation in Cartesian, Cylindrical, and Spherical Coordinate Systems Two-dimensional Steady Flow of Heat Circular Harmonics Conducting Cylinder in a Uniform Field General Cylindrical Harmonics Spherical Harmonics www.TechnicalBooksPDF.com coupled by mutual inductance, 197 discharge and charging of a capacitor, 194 electrical, 186 forced oscillations, 192 four-terminal networks, 208 attenuation and passbands, 214 table of transmission matrices, 211 filters, 289 Circuits: free oscillations, 190 general networks, 204 response to finite potential pulses, 203 steady state solutions, 207 Cofactor, 84 Collocation method, 583 Columns: buckling under axial load, 531 critical load, 532 Euler formula, 532 modes of buckling, 532 Complex impedance, 194 Complex integrals: branch points, 37 Bromwich, 34 definite, estimation of magnitude, indenting of contour, 33 Jordan’s lemma, 31 Complimentary error function, 41 Complimentary function, 55 Conformal representation, 361 modulus of the transformation, 362 Conjugate functions, 6, 359 application to hydrodynamic problems, 397 application to potential problems, 358 orthogonality, 359 Conservative systems, 245 small oscillations of, 247 Control systems, 321 Convolution theorem, 152 Coordinate systems: curvilinear, 918 curl, 921 divergence,919 gradient, 919 Coordinate systems: curvilinear: Laplacian, 920 orthogonal, 918 cylindrical, 921 generalized, 245 normal, 235, 239, 254 polar, 246 spherical polar, 922 Cord: arbitrary load, 520 concentrated load, 519 deflection of loaded, 516 on elastic support: concentrated load, 521 general load, 521 infinite with concentrated load, 522 uniform load, 522 uniform load, 517 over part of span, 518 Coriolis’ acceleration, 904 Cylindrical coordinates, 921 Cylindrical harmonics, general, 344 Damping matrix, 263 Degree of freedom, 232 Derivative of tabulated function, 277 Determinant, 83 cofactor, 84 Laplace’s expansion, 84 minor, 84 properties of, 86 Difference equation, 280 complimentary function, 280 homogeneous, 280 particular integral, 282 Difference table, 275 Differential equations, 53–81 approximate solutions, 577 Differential equations: complimentary function, 55 general solution, 55, 56 homogeneous first order, 54 linear, 53 nonlinear, 595–732 numerical solutions, 567 particular integral, 55 by complex algebra, 69 quasi-linear, 630 reduced equation, 56 solution by Laplace transform, 507 undetermined coefficients, 67 variable coefficients: comparison theorem, 72 Frobenius, method of, 73 indicial equation, 74 singular points, 73 variation of parameters, 75 Differentiation: of composite functions, 954 of definite integrals, 970 Diffusion equation, 412–34 Diffusivity, 418 Dirac delta function, 166 Dirichlet conditions, 857 Dissipation function, 263 Divergence theorem, 907 Duffing’s equation, 619 Dynamical matrix, 248 Dynamical system, 247 nonlinear, 604 Eigenfunction: of Sturm-Liouville differential equation, 77 of wave equation, 445 Eigenvalue problems, 78 Eigenvalues: matrix, 104 by matrix iteration, 252 nonconservative systems, 268 Sturm-Liouville differential equation, 77 wave equation, 445 Eigenvector expansion, 122 Eigenvectors: by matrix iteration, 252 of nonconservative system, 268 Electrical circuit (see Circuits) Electrostatics, 363, 934 capacitance of two eccentric cylinders (parallel line), 379 charge density, 394 conductor influenced by line charge, 388 conductors at different potentials, 388 effect of wall on uniform field, 394 field: around semi-infinite charged plate, 376 about spherical surface, 349 between concentric cylinders, 366 cylinder in uniform field, 342 external to elliptic cylinder, 371 between hyperbolic pole pieces, 373 between perpendicular semi-infinite plates, 373 in region around charged cylinder, 377 of a ring, 348 for surfaces expressed in parametric form, 382 Electrostatics: field: between two semi-infinite plates with slit, 372 parallel-plate capacitor, 389 surface charge, 363, 368 Elliptic integral, 846 Error constants: acceleration, 324 position, 323 velocity, 324 Error function, 41, 493, 814 Essential singularity, 19 Euler-Cauchy formula, 568 Euler formulas, Euler’s equation of motion, 925 Factorial function, 809 Feedback control systems, 321 Filter networks, 210, 289 associated matrix, 298 attenuation constant, 292 attenuation function, 214 characteristic impedance, 214, 294 cutoff frequency, 294 dissipationless, 215 high-pass, 294 low-pass, 294 mid-series terminations, 290 mid-shunt terminations, 290 passband, 215, 293 phase constant, 292 phase function, 214 propagation constant, 291 propagation function, 214 reflection coefficients, 296 stopband, 293 Final value theorem, 155 Finite differences, 273 Flexibility matrix, 257 Forced oscillations: linear systems, 192 nonlinear systems, 665 Four-terminal network, 208 Fourier integral, 869 Fourier-Mellin theorem, 148 Fourier series, 854, 857 complex form., 858 convergence, 864 expansion of functions, 860 real form, 859 Fourier transforms, 485, 872 heat conduction in semi-infinite slab, 539 infinite vibrating string, 541 Fourier’s law of heat conduction, 406 Free oscillations, 190 Frequency, fundamental, 249, 442 Functions: analytic, complex, line integrals of, composite, differentiation of, 959 conjugate, implicit, differentiation of, 959 linear dependence, 65 matrix, 116 meromorphic, 20 rms value, 865 Galerkin’s method, 590 Gamma function, 808 Gaussian (normal) distribution, 753 Gauss’s pi function, 809 Gauss’s theorem, 909 Generalized coordinates, 245 Generalized forces, 245 Geometric series, 820 Gradient, 906 Graeffe’s root-squaring method, 885 Gravitational potential, 929 Green’s functions, 306, 548 deflection of cantilever beam: arbitrary load, 551 with three point forces, 553 uniform load, 553 vibrations of a string with distributed forcing function, 549 Green’s theorem, 44, 910 Gregory-Newton interpolation, 276 Hamilton’s principle, 984 Hankel functions, 792 Hankel transform : definition, 485 vibration of infinite circular membrane, 545 Harmonic function: circular, 338 cylindrical, 344 degree of harmonic, 339 surface, 345 orthogonality, 347 zonal, 345, 347 Harmonic vibrations, 855 Harmonics, 443 Heat conduction, 412–434 in circular plate, 424 cooling of hot brick, 422 in cylinder, 340 electrical analogy, 416 of finite plate, 337 general theorems, 430 in infinite bar, 423 Heat conduction: in semi-infinite rod, 418 in semi-infinite solid, 491, 539, 546 with periodic surface temperature, 547 two-dimensional steady flow, 335 in two-dimensions, 419 variable linear flow, 413 Heaviside’s rules: expansion theorem, 168 second rule, 171 third rule, 171 Helmholtz’s equation, 474 Hill-Meissner equation, 135, 688 Hill’s equation, 132 Homogeneous equations: differential equations, 54 linear algebraic, 99, 102 Hydrodynamics, 397 Bernoulli’s equation, 926 complex potential, 398 continuity equation, 924 dividing streamline, 402 Euler’s equation, 925 flow out of a channel, 389 flow over blunt body, 402 flow over an obstacle, 394 infinite row of sources, 403 Joukowski transformation, 400 source-sink flow, 399 stagnation point, 402 stream function, 397 total fluid velocity, 401 uniform flow plus source, 401 uniform stream at angle of attack, 400 velocity potential, 397, 924 vortex flow, 400 Impedance: complex, 194 mechanical, 228, 269 Impedance matrix, 208 Implicit function, 961 Impulse response, 318 Inertia matrix, 248 Infinite series, 819 absolute convergence, 829 alternating series, 829 binomial, 843 Cauchy’s integral test, 826 Cauchy’s ratio test, 827 comparison test, 824 convergent and divergent, 821 geometric, 820 oscillating, 821 power series, 831 differentiation and integration of, 836, 840 equality, 837 evaluation of integrals, 846 interval of convergence, 831 product of two series, 833 quotient of two series, 834 reversion, 837 sum of two series, 832 sequence, 820 test for divergence, 825 uniform convergence, 838 Weierstrass M test, 839 Influence function, 306, 319 Inhomogeneous wave equation, 473 magnetic vector potential, 471 Initial value theorem, 156 Integral of a tabulated function, 277 Integral equations: Abel’s, 501 Integral equations: applied to time-varying electrical circuits, 505 of first kind reducible to equations with a difference kernel, 504 Volterra’s: of first kind with difference kernel, 501 of second kind with difference kernel, 499 Integration of definite integral, 972 Jacobi series, 798 Jacobi’s transformation, 140(prob 8) Jordan’s lemma, 31 Joukowski’s transformation, 400 Kinetic energy, 237, 245, 248 Kirchhoff’s laws, 187 Kryloff and Bogoliuboff, method of, 630 anharmonic oscillator, 632 van der Pol equation, 633 Lagrange multipliers, 968 Lagrange’s equations, 236, 984 Laplace transforms, 143–185 definition, 151, 485 deflection of loaded cord, 515 evaluation of integrals, 496 Fourier-Mellin theorem, 148 fractional powers of s, 159 linear heat flow in semi-infinite solid, 491 of linear integral equations, 499 oscillations of a bar, 494 Laplace transforms: of periodic functions, 172 simultaneous linear differential equations with constant coefficients, 178 solution for dissipation transmission line, 487 solution of ordinary differential equations with variable coefficients, 507 stretched cord with elastic supports, 521 to sum Fourier series, 511 theorems: complex differentiation, 157 complex shifting, 154 convolution (Faltung), 152 differentiation with respect to second independent variable, 158 final value, 155 initial value, 156 integration with respect to second independent variable, 158 real shifting, 154, 155 transforms of: constant, 151 definite integral, 156 first derivative, 151 general derivatives, 152 indefinite integral, 156 (See also Appendix A, Table 2) waves on infinite line, 489 Laplace’s equation, 333, 358 cartesian coordinates, 333 cylindrical coordinates, 333 spherical coordinates, 333 Laplace’s expansion, 84 Laurent series, 15 Least squares, 562 error of fit, 563 matrix formulation, 566 Legendre functions: Legendre coefficients, 804 Legendre’s differential equation, 799 Legendre’s polynomials, 800 associated, 807 generating function, 803 orthogonality, 805 Rodrigues’ formula, 801 second kind, 802 series expansions of, 807 L’Hospital’s rule, 851 Line integral, 912 complex, Linear algebraic equations, 98 consistent system., 98 Cramer’s rule, 101 homogeneous, 102 inconsistent system, 98 Linear differential operator, 57 Liouville-Neumann series, 500 Liouville’s theorem, 23 Logarithmic decrement, 227 Lorenz’s equation, 473 Maclaurin’s series, 842 Magnetostatics, 934 Marginal stability, 328 Mathieu-Hill equation, 681, 717 Matrices, 90–142 addition and subtraction, 91 adjoint, 94 adjoint method, 129 algebraic laws, 91 associate, 98 augmented, 99 Cayley-Hamilton theorem, 107 Matrices: characteristic equation, 104 characteristic matrix, 104 Choleski’s method, 113 commutable, 93 companion matrix, 137 conformable, 92 conjugate, 98 continued products, 93 diagonal, 90 differentiation, 120 eigenvalues, 104 eigenvector expansion, 122 eigenvectors, 105 functions of, 116 Hermitian, 98 imaginary, 98 integration, 120 inverse, 94 involutory, 98 matrizant, 125 modal, 122 multiplication, 92 null, 91 orthogonal, 98 partitioning, 97 Peano-Baker method, 125 postmultiplication, 93 power series, 116 premultiplication, 93 rank, 98 real, 98 reversal law, 95 scalar, 94 skew-Hermitian, 98 skew-symmetric, 91, 98 square, 90 Sylvester’s theorem, 114 symmetric, 91, 98 trace, 106 transpose, 90 Matrices: unit, 91 unitary, 98 Vandermonde, 118 Maxima and minima of functions, 963 of two variables, 965 Maxwell’s equations, 933 Mean, 734 Median, 737 Mellin transform: definition, 486 steady-state heat conduction in an infinite wedge, 543 Meromorphic function, 20 Mesh parameters, 205 Minor, 84 Mode, 736 Monte Carlo method, 573 Morera’s theorem, 47 Mutual inductance, 197 Mutual parameters, 205 Natural frequency, 191, 249 Networks, four-terminal, 208 Newton-Raphson method, 881 Nonlinear oscillations, 595–732 anharmonic oscillator, 632 aperiodic motion, 638 autonomous systems, 635 Brillouin-Wentzel-Kramer (BWK) method, 708, 717 Duffing’s equation, 619 electrical circuit, 611 forced, 665 nonlinear capacitor, 673 nonlinear inductor, 666 saturable reactor, 670 series connected magnetic amplifier, 676 Nonlinear oscillations: forced vibrations, 618 free pendulum, 598, 610 Hill-Meissner equation, 135, 688 jump phenomena, 621, 668 Kryloff-Bogoliuboff method, 630 anharmonic oscillator, 632 van der Pol equation, 633 oscillator, 614 damped by solid friction, 596 phase plane: separatrix, 642 singular points: focal, 639 nodal, 640 ordinary, 636 saddle, 638, 642 vortex, 638, 642 trajectory, 636 subharmonic response, 622 time-varying circuits, 699, 702 van der Pol equation, 633 Normal coordinates, 235, 239, 254 Normal (gaussian) distribution, 753 Normal modes, 251 Numerical integration of differential equations: Euler-Cauchy formula, 568 first-order, 567 higher-order, 571 Runge-Kutta method, 569 step size, 568 Operational calculus, 483–560 Orthogonal functions, 444 Oscillations of a bar, 494 Partial derivatives, 954 change of variables, 958 continuity, 954 Partial derivatives: differential of implicit functions, 961 maxima and minima, 963 total differential, 959 Partial fractions, 63 Particular integral, 55 Partitioning of matrices, 97 Peano-Baker method, 125 Perturbation method, 313 Pivotal expansion, 88 Poisson distribution, 751 Poisson’s equation, 363, 931 Polynomial equations, 879 Encke roots, 886 Graeffe’s root-squaring method, 885 complex roots, 890 repeated roots, 891 solution of cubic equation, 883 Population, 733 Power series (see Infinite series) Principal modes, 234 by matrix iteration, 252 orthogonality, 250 Principle of least action, 985 Probability: a posteriori, 737 a priori, 737 Bayes’ theorem, 740 bias, 742 binomial coefficient, 743 Chebyshev’s inequality, 749 compound events, 738 discrete distributions, 740 distributions: binomial, 750 continuous, 746 gaussian, 753 Poisson, 751 statistical, 733 exclusive events, 739 Probability: expectation, 747 function, 746 independent events, 739 permutations, 742 Stirling’s formula, 744 trial, 741 Random walk, 573, 575 Rayleigh equation, 648 Rayleigh-Ritz method, 587 Rayleigh’s method, 578 Relaxation oscillations, 634, 643, 651 Residue, 17 Residues, evaluation of: at multiple poles, 23 at simple poles, 22 Resonance, 193, 231 Reversal law of matrix products, 95 Reversion method, 653 Riemann-Lebesgue theorem, 864 Rms value, 865 Root locus plot, 330 Routh-Hurwitz stability criterion, 326 Runge-Kutta method, 569 Scalar, 894 Schwarz’s transformation, 383 polygon with one angle, 386 polygon with zero angle, 386 successive transformations, 387 Second moment, 735, 747 Separation of variables, 336 Sequence, 820 Series (see Infinite series) Simpson’s rule, 278 Singular points, 19 at infinity, 20 order, 19 poles, 19 Skin effect, 419, 427, 936 on plane surface, 426 Spherical polar coordinates, 922 Stability: control system., 328 general system, 325, 326 Standard deviation, 736, 747 of the mean, 759 Steady heat flow, 404 conduction in an annular segment, 404 in an elliptical slab, 407 Steady state, 193 Stiffness matrix, 248 Stirling’s formula, 744 Stoke’s theorem, 917 Stream function, 397 Sturm-Liouville differential equation, 77 Sylvester’s theorem, 114 Tautochrone, 502 Taylor’s series, 841 from Cauchy’s integral formula, 14 function of n variables, 845 function of two variables, 845 symbolic form, 844 Tensor analysis, 936–951 addition, multiplication, and contraction, 942 applications: Lagrange’s equations, 948 particle dynamics, 948 work and energy, 947 associated tensors, 943 Tensor analysis: Christoffel symbols, 946 coordinate transformation, 937 differentiation: of an invariant, 945 of a tensor, 946, 947 Kronecker delta, 938 scalar, 940 summation convention, 938 vectors, contravariant and co-variant, 940 Three-eighths rule of Cotes, 279 Total differential, 959 Transcendental equations, 879 graphical solution, 880 Newton-Raphson method, 881 Transfer function, 306, 307 closed-loop, 322 combining, 309 feedback, 322 forward, 322 by matrix method, 312 open-loop, 322 by perturbation method, 313 of simple systems, 306 Transforms: Fourier, 485, 872 Hankel, 485 Laplace, 143–185, 485 Mellin, 486 multiple, 546 Transmission line, 215 angle of line, 217 attenuation function, 216 characteristic impedance, 216 with discrete leaks, 287 dissipationless, 483 distortionless, 458 equation of, 456 without leakage, 459 Transmission line: phase function, 216 propagation function, 216 Transmission matrix, 209 of four-terminal structures, table, 211 Undetermined coefficients, method of, 67 Unstable system, 326 Vandermonde matrix, 118 van der Pol’s equation, 633, 645, 652 Variance, 736, 747 Variate, 734 Variation of parameters, 75 Vector analysis, 894–934 addition and subtraction, 895 application to heat flow, 927 application to hydrodynamics, 923 cross product, 898 curl, 913 derivatives of vectors, 902 divergence theorem, 907 equality of vectors, 896 Gauss’s theorem, 909 gradient, 906 gravitational potential, 929 Green’s theorem, 910 line integral, 911 Maxwell’s equations, 933 multiple products, 900 orthogonal coordinates, 918 cylindrical, 921 spherical, 922 Vector analysis: Poisson’s equation, 931 scalar product, 897 scalar triple product, 901 Stoke’s theorem, 917 unit vector, 896 Velocity potential, 397 Vibrations : critically damped, 191 oscillatory, 191 overdamped, 191 Volterra’s integral equations: of first kind with difference kernel, 501 of second kind with difference kernel, 499 Wave equation, 435, 935 D’Alembert’s solution, 437 harmonic waves, 439 Helmholtz equation, 474 inhomogeneous, 470 orthogonal functions, 444 oscillations of hanging chain, 446 plucked string, 443 retarded potentials, 475 Wave equation: sound waves, 467 adiabatic, 469 isothermal, 468 stationary wave, 442 transmission-line, 456 distortionless, 458 line without leakage, 459 transverse vibrations of stretched string, 436 vibrations of circular membrane, 453 vibrations of rectangular membrane, 449 wave number, 439 waveguides, 476 waves in a canal, 464 Wave number, 439 Wave propagation, 868 plane wave, 869 wavelength, 869 Waveguides, 476 propagation constant, 476 TE waves, 477 TM waves, 479 Weierstrass M test, 839 Weighting function, 319 Wronskian, 65, 133 www.doverpublications.com .. .Applied Mathematics for Engineers and Physicists Third Edition Louis A Pipes and Lawrence R Harvill DOVER PUBLICATIONS, INC MINEOLA, NEW YORK www.TechnicalBooksPDF.com Copyright... www.doverpublications.com www.TechnicalBooksPDF.com PREFACE TO THE DOVER EDITION This book represents the reprinting of the third edition of Applied Mathematics for Engineers and Physicists, originally published... specially for this Dover edition Library of Congress Cataloging-in-Publication Data Pipes, Louis Albert, 1910– author Applied mathematics for engineers and physicists / Louis A Pipes and Lawrence