Mathematical Methods for Physicists

by Arfken; Weber

• ISBN13: 9780120598762
• ISBN10: 0120598760
• Edition: 6th
• Format: Hardcover
• Copyright: 2005-06-21
• Publisher: Elsevier Science & Technology

Summary

This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition. * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted

Table of Contents

Preface	xi
Vector Analysis	1	(102)
Definitions, Elementary Approach	1	(6)
Rotation of the Coordinate Axes	7	(5)
Scalar or Dot Product	12	(6)
Vector or Cross Product	18	(7)
Triple Scalar Product, Triple Vector Product	25	(7)
Gradient	32	(6)
Divergence	38	(5)
Curl, x	43	(6)
Successive Applications	49	(5)
Vector Integration	54	(6)
Gauss' Theorem	60	(4)
Stokes' Theorem	64	(4)
Potential Theory	68	(11)
Gauss' Law, Poisson's Equation	79	(4)
Dirac Delta Function	83	(12)
Helmholtz's Theorem	95	(8)
Additional Readings	101	(2)
Vector Analysis in Curved Coordinates and Tensors	103	(62)
Orthogonal Coordinates in R3	103	(7)
Differential Vector Operators	110	(4)
Special Coordinate Systems: Introduction	114	(1)
Circular Cylinder Coordinates	115	(8)
Spherical Polar Coordinates	123	(10)
Tensor Analysis	133	(6)
Contraction, Direct Product	139	(2)
Quotient Rule	141	(1)
Pseudotensors, Dual Tensors	142	(9)
General Tensors	151	(9)
Tensor Derivative Operators	160	(5)
Additional Readings	163	(2)
Determinants and Matrices	165	(76)
Determinants	165	(11)
Matrices	176	(19)
Orthogonal Matrices	195	(13)
Hermitian Matrices, Unitary Matrices	208	(7)
Diagonalization of Matrices	215	(16)
Normal Matrices	231	(10)
Additional Readings	239	(2)
Group Theory	241	(80)
Introduction to Group Theory	241	(5)
Generators of Continuous Groups	246	(15)
Orbital Angular Momentum	261	(5)
Angular Momentum Coupling	266	(12)
Homogeneous Lorentz Group	278	(5)
Lorentz Covariance of Maxwell's Equations	283	(8)
Discrete Groups	291	(13)
Differential Forms	304	(17)
Additional Readings	319	(2)
Infinite Series	321	(82)
Fundamental Concepts	321	(4)
Convergence Tests	325	(14)
Alternating Series	339	(3)
Algebra of Series	342	(6)
Series of Functions	348	(4)
Taylor's Expansion	352	(11)
Power Series	363	(7)
Elliptic Integrals	370	(6)
Bernoulli Numbers, Euler--Maclaurin Formula	376	(13)
Asymptotic Series	389	(7)
Infinite Products	396	(7)
Additional Readings	401	(2)
Functions of a Complex Variable I Analytic Properties, Mapping	403	(52)
Complex Algebra	404	(9)
Cauchy--Riemann Conditions	413	(5)
Cauchy's Integral Theorem	418	(7)
Cauchy's Integral Formula	425	(5)
Laurent Expansion	430	(8)
Singularities	438	(5)
Mapping	443	(8)
Conformal Mapping	451	(4)
Additional Readings	453	(2)
Functions of a Complex Variable II	455	(44)
Calculus of Residues	455	(27)
Dispersion Relations	482	(7)
Method of Steepest Descents	489	(10)
Additional Readings	497	(2)
The Gamma Function (Factorial Function)	499	(36)
Definitions, Simple Properties	499	(11)
Digamma and Polygamma Functions	510	(6)
Stirling's Series	516	(4)
The Beta Function	520	(7)
Incomplete Gamma Function	527	(8)
Additional Readings	533	(2)
Differential Equations	535	(86)
Partial Differential Equations	535	(8)
First-Order Differential Equations	543	(11)
Separation of Variables	554	(8)
Singular Points	562	(3)
Series Solutions---Frobenius' Method	565	(13)
A Second Solution	578	(14)
Nonhomogeneous Equation---Green's Function	592	(19)
Heat Flow, or Diffusion, PDE	611	(10)
Additional Readings	618	(3)
Sturm--Liouville Theory---Orthogonal Functions	621	(54)
Self-Adjoint ODEs	622	(12)
Hermitian Operators	634	(8)
Gram-Schmidt Orthogonalization	642	(7)
Completeness of Eigenfunctions	649	(13)
Green's Function---Eigenfunction Expansion	662	(13)
Additional Readings	674	(1)
Bessel Functions	675	(66)
Bessel Functions of the First Kind, Jv(x)	675	(19)
Orthogonality	694	(5)
Neumann Functions	699	(8)
Hankel Functions	707	(6)
Modified Bessel Functions, Iv(x) and Kv(x)	713	(6)
Asymptotic Expansions	719	(6)
Spherical Bessel Functions	725	(16)
Additional Readings	739	(2)
Legendre Functions	741	(76)
Generating Function	741	(8)
Recurrence Relations	749	(7)
Orthogonality	756	(11)
Alternate Definitions	767	(4)
Associated Legendre Functions	771	(15)
Spherical Harmonics	786	(7)
Orbital Angular Momentum Operators	793	(4)
Addition Theorem for Spherical Harmonics	797	(6)
Integrals of Three Y's	803	(3)
Legendre Functions of the Second Kind	806	(7)
Vector Spherical Harmonics	813	(4)
Additional Readings	816	(1)
More Special Functions	817	(64)
Hermite Functions	817	(20)
Laguerre Functions	837	(11)
Chebyshev Polynomials	848	(11)
Hypergeometric Functions	859	(4)
Confluent Hypergeometric Functions	863	(6)
Mathieu Functions	869	(12)
Additional Readings	879	(2)
Fourier Series	881	(50)
General Properties	881	(7)
Advantages, Uses of Fourier Series	888	(4)
Applications of Fourier Series	892	(11)
Properties of Fourier Series	903	(7)
Gibbs Phenomenon	910	(4)
Discrete Fourier Transform	914	(5)
Fourier Expansions of Mathieu Functions	919	(12)
Additional Readings	929	(2)
Integral Transforms	931	(74)
Integral Transforms	931	(5)
Development of the Fourier Integral	936	(2)
Fourier Transforms---Inversion Theorem	938	(8)
Fourier Transform of Derivatives	946	(5)
Convolution Theorem	951	(4)
Momentum Representation	955	(6)
Transfer Functions	961	(4)
Laplace Transforms	965	(6)
Laplace Transform of Derivatives	971	(8)
Other Properties	979	(11)
Convolution (Faltungs) Theorem	990	(4)
Inverse Laplace Transform	994	(11)
Additional Readings	1003	(2)
Integral Equations	1005	(32)
Introduction	1005	(7)
Integral Transforms, Generating Functions	1012	(6)
Neumann Series, Separable (Degenerate) Kernels	1018	(11)
Hilbert--Schmidt Theory	1029	(8)
Additional Readings	1036	(1)
Calculus of Variations	1037	(42)
A Dependent and an Independent Variable	1038	(6)
Applications of the Euler Equation	1044	(8)
Several Dependent Variables	1052	(4)
Several Independent Variables	1056	(2)
Several Dependent and Independent Variables	1058	(2)
Lagrangian Multipliers	1060	(5)
Variation with Constraints	1065	(7)
Rayleigh--Ritz Variational Technique	1072	(7)
Additional Readings	1076	(3)
Nonlinear Methods and Chaos	1079	(30)
Introduction	1079	(1)
The Logistic Map	1080	(5)
Sensitivity to Initial Conditions and Parameters	1085	(3)
Nonlinear Differential Equations	1088	(21)
Additional Readings	1107	(2)
Probability	1109	(44)
Definitions, Simple Properties	1109	(7)
Random Variables	1116	(12)
Binomial Distribution	1128	(2)
Poisson Distribution	1130	(4)
Gauss' Normal Distribution	1134	(4)
Statistics	1138	(15)
Additional Readings	1150	(1)
General References	1150	(3)
Index	1153