Mathematical Methods for Physics and Engineering

1 This page intentionally left blankMathematical Methods for Physics and EngineeringThe third edition of this highly acclaimed undergraduate textbook is suitablefor teaching all the mathematics ever likely to be needed for an undergraduatecourse in any of the physical sciences. As well as lucid descriptions of all thetopics covered and many worked examples, it contains more than 800 number of additional topics have been included and the text has undergonesignificant reorganisation in some areas. New stand-alone chapters: give a systematic account of the special functions of physical science cover an extended range of practical applications of complex variables includingWKB Methods and saddle-point integration techniques provide an introduction to quantum tabulations, of relevance in statistics and numerical integration, havebeen added.

2 In this edition, all 400 odd-numbered exercises are provided withcomplete worked solutions in a separate manual, available to both students andtheir teachers; these are in addition to the hints and outline answers given inthe main text. The even-numbered exercises have no hints, answers or workedsolutions and can be used for unaided homework; full solutions to them areavailable to instructors on a password-protected Rileyread mathematics at the University of Cambridge and proceededto a there in theoretical and experimental nuclear Physics . He became aresearch associate in elementary particle Physics at Brookhaven, and then, havingtaken up a lectureship at the Cavendish Laboratory, Cambridge, continued thisresearch at the Rutherford Laboratory and Stanford; in particular he was involvedin the experimental discovery of a number of the early baryonic resonances.

3 Aswell as having been Senior Tutor at Clare college , where he has taught physicsand mathematics for over 40 years, he has served on many committees concernedwith the teaching and examining of these subjects at all levels of tertiary andundergraduate education. He is also one of the authors of200 Puzzling Hobsonread natural sciences at the University of Cambridge, spe-cialising in theoretical Physics , and remained at the Cavendish Laboratory tocomplete a in the Physics of star-formation. As a research fellow at TrinityHall, Cambridge and subsequently an advanced fellow of the Particle Physicsand Astronomy Research Council, he developed an interest in cosmology, andin particular in the study of fluctuations in the cosmic microwave was involved in the first detection of these fluctuations using a ground-basedinterferometer.

4 He is currently a University Reader at the Cavendish Laboratory,his research interests include both theoretical and observational aspects of cos-mology, and he is the principal author ofGeneral Relativity: An Introduction forPhysicists. He is also a Director of Studies in Natural Sciences at Trinity Hall andenjoys an active role in the teaching of undergraduate Physics and Benceobtained both his undergraduate degree in Natural Sciencesand his in Astrophysics from the University of Cambridge. He then becamea Research Associate with a special interest in star-formation processes and thestructure of star-forming regions. In particular, his research concentrated on thephysics of jets and outflows from young stars. He has had considerable experi-ence of teaching mathematics and Physics to undergraduate and Methodsfor Physics and EngineeringThird RILEY, HOBSON and BENCE cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S o PauloCambridge University PressThe Edinburgh Building, Cambridgecb2 2ru,UKFirst published in print formatisbn-13978-0-521-86153-3isbn-13978 -0-521-67971-8isbn-13978-0-511-16842-0 K.

5 F. Riley, M. P. Hobson and S. J. Bence 20062006 publication is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University University Press has no responsibility for the persistence or accuracy ofurlsfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or in the United States of America by Cambridge University Press, New (EBL)eBook (EBL)hardbackContentsPreface to the third editionpagexxPreface to the second editionxxiiiPreface to the first editionxxv1 Preliminary functions and equations1 Polynomial equations; factorisation; properties of identities10 Single angle; compound angles.

6 Double- and half-angle fractions18 Complications and special of binomial particular Methods of proof30 Proof by induction; proof by contradiction; necessary and sufficient and answers392 Preliminary from first principles; products; the chain rule; quotients;implicit differentiation; logarithmic differentiation; Leibnitz theorem; specialpoints of a function; curvature; theorems of from first principles; the inverse of differentiation; by inspec-tion; sinusoidal functions; logarithmic integration; using partial fractions;substitution method; integration by parts; reduction formulae; infinite andimproper integrals; plane polar coordinates; integral inequalities; applicationsof and answers813 Complex numbers and hyperbolic need for complex of complex numbers85 Addition and subtraction; modulus and argument; multiplication; complexconjugate; representation of complex numbers92 Multiplication and division in polar Moivre s theorem95trigonometric identities; finding thenth roots of unity; solving logarithms and complex to differentiation and functions102 Definitions; hyperbolic trigonometric analogies; identities of hyperbolicfunctions; solving hyperbolic equations; inverses of hyperbolic functions;calculus of hyperbolic and answers1134 Series and of series116 Arithmetic series; geometric series; arithmetico-geometric series; the differencemethod; series involving natural numbers; transformation of of infinite series124 Absolute and conditional convergence; series containing only real positiveterms; alternating series with series131 Convergence of power series.

7 Operations with power series136 Taylor s theorem; approximation errors; standard Maclaurin of and answers149viCONTENTS5 Partial of the partial total differential and total and inexact theorems of partial chain of s theorem for many-variable values of many-variable values under Thermodynamic Differentiation of Hints and answers1856 Multiple of multiple integrals191 Areas and volumes; masses, centres of mass and centroids; Pappus theorems;moments of inertia; mean values of of variables in multiple integrals199 Change of variables in double integrals; evaluation of the integralI= e x2dx; change of variables in triple integrals; general properties and answers2117 Vector and and subtraction of by a vectors and of a of vectors219 Scalar product; vector product; scalar triple product; vector triple of lines, planes and vectors to find distances229 Point to line; point to plane; line to line; line to Hints and answers2408 Matrices and vector spaces242 Basis vectors; inner product; some useful matrix algebra250 Matrix addition; multiplication by a scalar; matrix of transpose of a complex and Hermitian conjugates of a trace of a determinant of a matrix259 Properties of The inverse of a The rank of a Special types of square matrix268 Diagonal; triangular; symmetric andantisymmetric; orthogonal; Hermitianand anti-Hermitian; unitary; Eigenvectors and eigenvalues272Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitarymatrix.

8 Of a general square Determination of eigenvalues and eigenvectors280 Degenerate Change of basis and similarity Diagonalisation of Quadratic and Hermitian forms288 Stationary properties of the eigenvectors; quadratic Simultaneous linear equations292 Range; null space;Nsimultaneous linear equations inNunknowns; singularvalue Hints and answers3149 Normal oscillatory and normal Ritz and answers33210 Vector Differentiation of vectors334 Composite vector expressions; differential of a Integration of Space Vector functions of several Scalar and vector Vector operators347 Gradient of a scalar field; divergence of a vector field; curl of a vector Vector operator formulae354 Vector operators acting on sums and products; combinations of grad, div Cylindrical and spherical polar General curvilinear Hints and answers37511 Line, surface and volume Line integrals377 Evaluating line integrals; physical examples; line integrals with respect to Connectivity of Green s theorem in a Conservative fields and Surface integrals389 Evaluating surface integrals; vector areas of surfaces; physical Volume integrals396 Volumes of three-dimensional Integral forms for grad, div and Divergence theorem and related theorems401 Green s theorems; other related integral theorems; physical Stokes theorem and related theorems406 Related integral theorems.

9 Physical Hints and answers41412 Fourier The Dirichlet The Fourier Symmetry Discontinuous Non-periodic Integration and Complex Fourier Parseval s Hints and answers43113 Integral Fourier transforms433 The uncertainty principle; Fraunhofer diffraction; the Dirac -function;relation of the -function to Fourier transforms; properties of Fouriertransforms; odd and even functions; convolution and deconvolution; correlationfunctions and energy spectra; Parseval s theorem; Fourier transforms in Laplace transforms453 Laplace transforms of derivatives and integrals; other properties of Concluding Hints and answers46614 First-order ordinary differential General form of First-degree first-order equations470 Separable-variable equations; exact equations; inexact equations, integrat-ing factors; linear equations; homogeneous equations; isobaric equations;Bernoulli s equation; miscellaneous Higher-degree first-order equations480 Equations soluble forp;forx;fory; Clairaut s Hints and answers48815 Higher-order ordinary differential Linear equations with constant coefficients492 Finding the complementary functionyc(x); finding the particular integralyp(x); constructing the general solutionyc(x)+yp(x); linear recurrencerelations; Laplace transform Linear equations with variable coefficients503 The Legendre and Euler linear equations; exact equations; partially knowncomplementary function; variation of parameters; Green s functions.

10 Canonicalform for second-order General ordinary differential equations518 Dependent variable absent; independent variable absent; non-linear exactequations; isobaric or homogeneous equations; equations homogeneous inxoryalone; equations havingy=Aexas a Hints and answers52916 Series solutions of ordinary differential Second-order linear ordinary differential equations531 Ordinary and singular Series solutions about an ordinary Series solutions about a regular singular point538 Distinct roots not differing by an integer; repeated root of the indicialequation; distinct roots differing by an Obtaining a second solution544 The Wronskian method; the derivative method; series form of the Polynomial Hints and answers55317 Eigenfunction Methods for differential Sets of functions556 Some useful Adjoint, self-adjoint and Hermitian Properties of Hermitian operators561 Reality of the eigenvalues; orthogonality of the eigenfunctions; constructionof real Sturm Liouville equations564 Valid boundary conditions; putting an equation into Sturm Liouville Superposition of eigenfunctions: Green s A useful Hints and answers57618 Special Legendre functions577 General solution for integer ; properties of Legendre Associated Legendre Spherical Chebyshev Bessel functions602 General solution for non-integer ; general solution for integer.