Mathematical methods for physicists : a comprehensive guide

Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises. Revised and updated version of the leading text in mathematical physics Focuses on problem-solving skills and active learning, offering numerous chapter problems Clearly identified definitions, theorems, and proofs promote clarity and understanding New to this edition: Improved modular chapters New up-to-date examples More intuitive explanations 1.1......Page 2 Mathematical Methods for Physicists: A Comprehensive Guide......Page 3 Copyright......Page 4 00 cover.pdf......Page 0 To the Student......Page 11 What's New......Page 12 Acknowledgments......Page 13 1.1 Infinite Series......Page 14 Fundamental Concepts......Page 15 Comparison Test......Page 16 D'Alembert (or Cauchy) Ratio Test......Page 17 Cauchy (or Maclaurin) Integral Test......Page 18 More Sensitive Tests......Page 21 Alternating Series......Page 24 Absolute and Conditional Convergence......Page 26 Operations on Series......Page 27 Improvement of Convergence......Page 29 Rearrangement of Double Series......Page 31 Uniform Convergence......Page 34 Weierstrass M (Majorant) Test......Page 35 Abel's Test......Page 36 Properties of Uniformly Convergent Series......Page 37 Taylor's Expansion......Page 38 Power Series......Page 40 Properties of Power Series......Page 42 Uniqueness Theorem......Page 43 Indeterminate Forms......Page 44 Inversion of Power Series......Page 45 1.3 Binomial Theorem......Page 46 1.4 Mathematical Induction......Page 53 1.5 Operations on Series Expansions of Functions......Page 54 1.6 Some Important Series......Page 58 1.7 Vectors......Page 59 Basic Properties......Page 60 Dot (Scalar) Product......Page 62 Orthogonality......Page 64 Basic Properties......Page 66 Functions in the Complex Domain......Page 68 Polar Representation......Page 69 Complex Numbers of Unit Magnitude......Page 70 Circular and Hyperbolic Functions......Page 71 Powers and Roots......Page 72 Logarithm......Page 73 1.9 Derivatives and Extrema......Page 75 Stationary Points......Page 76 Integration by Parts......Page 78 Special Functions......Page 79 Other Methods......Page 80 Multiple Integrals......Page 83 Remarks: Changes of Integration Variables......Page 85 1.11 Dirac Delta Function......Page 88 Properties of δ(x)......Page 91 Kronecker Delta......Page 92 Additional Readings......Page 95 Homogeneous Linear Equations......Page 96 Inhomogeneous Linear Equations......Page 97 Definitions......Page 98 Properties of Determinants......Page 100 Linear Equation Systems......Page 101 Determinants and Linear Dependence......Page 102 Linearly Dependent Equations......Page 103 Numerical Evaluation......Page 104 Basic Definitions......Page 108 Addition, Subtraction......Page 109 Matrix Multiplication (Inner Product)......Page 110 Matrix Inverse......Page 112 Systems of Linear Equations......Page 115 Determinant Product Theorem......Page 116 Transpose, Adjoint, Trace......Page 117 Matrix Representation of Vectors......Page 119 Unitary Matrices......Page 120 Direct Product......Page 121 Functions of Matrices......Page 126 Additional Readings......Page 134 Vector Analysis......Page 135 3.1 Review of Basic Properties......Page 136 Vector or Cross Product......Page 138 Scalar Triple Product......Page 140 Vector Triple Product......Page 142 Rotations......Page 145 Orthogonal Transformations......Page 147 Reflections......Page 148 Successive Operations......Page 149 3.4 Rotations in R3......Page 151 Gradient, ∇......Page 155 Divergence, ∇·......Page 158 Curl, ∇×......Page 161 Successive Applications of ∇......Page 165 Irrotational and Solenoidal Vector Fields......Page 166 Vector Laplacian......Page 167 Miscellaneous Vector Identities......Page 168 Line Integrals......Page 171 Surface Integrals......Page 173 Volume Integrals......Page 174 Gauss' Theorem......Page 176 Green's Theorem......Page 177 Stokes' Theorem......Page 179 3.9 Potential Theory......Page 182 Scalar Potential......Page 183 Vector Potential......Page 184 Gauss' Law......Page 187 Helmholtz's Theorem......Page 189 Orthogonal Coordinates in R3......Page 194 Integrals in Curvilinear Coordinates......Page 196 Differential Operators in Curvilinear Coordinates......Page 197 Circular Cylindrical Coordinates......Page 199 Spherical Polar Coordinates......Page 202 Rotation and Reflection in Spherical Coordinates......Page 207 Additional Readings......Page 215 Introduction, Properties......Page 216 Covariant and Contravariant Tensors......Page 217 Tensors of Rank 2......Page 218 Symmetry......Page 219 Contraction......Page 220 Direct Product......Page 221 Quotient Rule......Page 222 Spinors......Page 224 Pseudotensors......Page 226 Dual Tensors......Page 227 Metric Tensor......Page 229 Covariant and Contravariant Bases......Page 231 Covariant Derivatives......Page 233 Evaluating Christoffel Symbols......Page 234 Tensor Derivative Operators......Page 235 4.4 Jacobians......Page 238 Inverse of Jacobian......Page 241 Introduction......Page 243 Exterior Algebra......Page 245 Complementary Differential Forms......Page 246 Exterior Derivatives......Page 249 4.7 Integrating Forms......Page 254 Stokes' Theorem......Page 256 Additional Readings......Page 260 5.1 Vectors in Function Spaces......Page 261 Scalar Product......Page 264 Hilbert Space......Page 265 Orthogonal Expansions......Page 267 Expansions and Scalar Products......Page 270 Bessel's Inequality......Page 272 Expansions of Dirac Delta Function......Page 273 Dirac Notation......Page 275 5.2 Gram-Schmidt Orthogonalization......Page 279 Orthonormalizing Physical Vectors......Page 282 5.3 Operators......Page 285 Commutation of Operators......Page 286 Identity, Inverse, Adjoint......Page 287 Basis Expansions of Operators......Page 289 Basis Expansion of Adjoint......Page 291 Functions of Operators......Page 292 5.4 Self-Adjoint Operators......Page 293 Unitary Transformations......Page 297 Successive Transformations......Page 300 5.6 Transformations of Operators......Page 302 Nonunitary Transformations......Page 303 5.7 Invariants......Page 304 5.8 Summary—Vector Space Notation......Page 306 Additional Readings......Page 307 6.1 Eigenvalue Equations......Page 308 Equivalence of Operator and Matrix Forms......Page 309 A Preliminary Example......Page 310 Another Eigenproblem......Page 314 Degeneracy......Page 316 6.3 Hermitian Eigenvalue Problems......Page 319 6.4 Hermitian Matrix Diagonalization......Page 320 Finding a Diagonalizing Transformation......Page 322 Simultaneous Diagonalization......Page 323 Spectral Decomposition......Page 324 Expectation Values......Page 325 Positive Definite and Singular Operators......Page 326 6.5 Normal Matrices......Page 328 Nonnormal Matrices......Page 331 Defective Matrices......Page 333 Additional Readings......Page 337 7.1 Introduction......Page 338 Separable Equations......Page 340 Exact Differentials......Page 342 Equations Homogeneous in x and y......Page 343 Isobaric Equations......Page 344 Linear First-Order ODEs......Page 345 7.3 ODEs with Constant Coefficients......Page 351 Singular Points......Page 352 7.5 Series Solutions—Frobenius' Method......Page 355 First Example—Linear Oscillator......Page 356 Expansion about x0......Page 359 A Second Example—Bessel's Equation......Page 360 Regular and Irregular Singularities......Page 362 Summary......Page 364 7.6 Other Solutions......Page 367 Linear Independence of Solutions......Page 368 Number of Solutions......Page 370 Finding a Second Solution......Page 371 Series Form of the Second Solution......Page 373 Summary......Page 378 Variation of Parameters......Page 384 7.8 Nonlinear Differential Equations......Page 386 Fixed and Movable Singularities, Special Solutions......Page 387 Additional Readings......Page 389 8.1 Introduction......Page 390 Self-Adjoint ODEs......Page 393 Making an ODE Self-Adjoint......Page 394 8.3 ODE Eigenvalue Problems......Page 398 8.4 Variation Method......Page 404 8.5 Summary, Eigenvalue Problems......Page 407 Additional Readings......Page 408 9.1 Introduction......Page 409 Examples of PDEs......Page 410 9.2 First-Order Equations......Page 411 Characteristics......Page 412 More General PDEs......Page 414 More Than Two Independent Variables......Page 415 Classes of PDEs......Page 417 Boundary Conditions......Page 419 Nonlinear PDEs......Page 421 9.4 Separation of Variables......Page 422 Cartesian Coordinates......Page 423 Circular Cylindrical Coordinates......Page 429 Spherical Polar Coordinates......Page 432 Summary: Separated-Variable Solutions......Page 438 9.5 Laplace and Poisson Equations......Page 441 9.6 Wave Equation......Page 443 d'Alembert's Solution......Page 444 9.7 Heat-Flow, or Diffusion PDE......Page 445 Alternate Solutions......Page 447 9.8 Summary......Page 452 Additional Readings......Page 453 10 Green's Functions......Page 454 10.1 One-Dimensional Problems......Page 455 General Properties......Page 456 Form of Green's Function......Page 457 Other Boundary Conditions......Page 459 Relation to Integral Equations......Page 462 Basic Features......Page 466 Eigenfunction Expansions......Page 467 Form of Green's Functions......Page 468 Additional Readings......Page 474 11 Complex Variable Theory......Page 475 11.1 Complex Variables and Functions......Page 476 11.2 Cauchy-Riemann Conditions......Page 477 Analytic Functions......Page 478 Derivatives of Analytic Functions......Page 480 Point at Infinity......Page 481 Contour Integrals......Page 483 Statement of Theorem......Page 484 Cauchy's Theorem: Proof......Page 487 Multiply Connected Regions......Page 489 11.4 Cauchy's Integral Formula......Page 492 Derivatives......Page 494 Morera's Theorem......Page 495 Further Applications......Page 496 Taylor Expansion......Page 498 Laurent Series......Page 500 Poles......Page 503 Branch Points......Page 505 Analytic Continuation......Page 509 Residue Theorem......Page 515 Computing Residues......Page 516 Cauchy Principal Value......Page 518 Pole Expansion of Meromorphic Functions......Page 521 Counting Poles and Zeros......Page 524 Product Expansion of Entire Functions......Page 525 Trigonometric Integrals, Range (0,2π)......Page 528 Integrals, Range -∞ to ∞......Page 531 Integrals with Complex Exponentials......Page 533 Another Integration Technique......Page 537 Avoidance of Branch Points......Page 538 Exploiting Branch Cuts......Page 540 Exploiting Periodicity......Page 543 11.9 Evaluation of Sums......Page 550 Mapping......Page 553 Additional Readings......Page 556 Rodrigues Formulas......Page 557 Schlaefli Integral......Page 560 Generating Functions......Page 561 Finding Generating Functions......Page 562 Summary—Orthogonal Polynomials......Page 564 12.2 Bernoulli Numbers......Page 566 Bernoulli Polynomials......Page 571 12.3 Euler-Maclaurin Integration Formula......Page 573 12.4 Dirichlet Series......Page 577 12.5 Infinite Products......Page 580 12.6 Asymptotic Series......Page 583 Exponential Integral......Page 584 Cosine and Sine Integrals......Page 587 Definition of Asymptotic Series......Page 589 12.7 Method of Steepest Descents......Page 591 Saddle Points......Page 592 Saddle Point Method......Page 594 12.8 Dispersion Relations......Page 597 Symmetry Relations......Page 599 Optical Dispersion......Page 600 The Parseval Relation......Page 601 Additional Readings......Page 604 Infinite Limit (Euler)......Page 605 Definite Integral (Euler)......Page 606 Infinite Product (Weierstrass)......Page 608 Functional Relations......Page 609 Schlaefli Integral......Page 610 Factorial Notation......Page 612 Digamma Function......Page 616 Polygamma Function......Page 618 Series Summation......Page 619 13.3 The Beta Function......Page 623 Derivation of Legendre Duplication Formula......Page 624 13.4 Stirling's Series......Page 628 Derivation from Euler-Maclaurin Integration Formula......Page 629 Stirling’s Formula......Page 630 13.5 Riemann Zeta Function......Page 632 Incomplete Gamma Functions......Page 639 Exponential Integral......Page 640 Error Function......Page 643 Additional Readings......Page 647 14.1 Bessel Functions of the First Kind, Jν(x)......Page 648 Generating Function for Integral Order......Page 649 Recurrence Relations......Page 650 Bessel's Differential Equation......Page 651 Integral Representation......Page 652 Zeros of Bessel Functions......Page 653 Schlaefli Integral......Page 658 14.2 Orthogonality......Page 666 Normalization......Page 667 Bessel Series......Page 668 Definition and Series Form......Page 672 Recurrence Relations......Page 674 Wronskian Formulas......Page 675 Uses of Neumann Functions......Page 676 14.4 Hankel Functions......Page 679 Definitions......Page 680 Contour Integral Representation of the Hankel Functions......Page 681 14.5 Modified Bessel Functions, Iν(x) and Kν(x)......Page 685 Recurrence Relations for Iν......Page 686 Second Solution Kν......Page 687 Integral Representations......Page 688 Summary......Page 690 Asymptotic Forms of Hankel Functions......Page 693 Expansion of an Integral Representation for Kν......Page 695 Additional Asymptotic Forms......Page 697 Properties of the Asymptotic Forms......Page 698 14.7 Spherical Bessel Functions......Page 703 Definitions......Page 704 Recurrence Relations......Page 707 Limiting Values......Page 708 Orthogonality and Zeros......Page 709 Modifed Spherical Bessel Functions......Page 711 Additional Readings......Page 718 15 Legendre Functions......Page 719 15.1 Legendre Polynomials......Page 720 Recurrence Formulas......Page 722 Rodrigues Formula......Page 724 15.2 Orthogonality......Page 728 Legendre Series......Page 730 15.3 Physical Interpretation of Generating Function......Page 740 Electric Multipoles......Page 741 15.4 Associated Legendre Equation......Page 745 Associated Legendre Polynomials......Page 747 Associated Legendre Functions......Page 748 Orthogonality......Page 750 15.5 Spherical Harmonics......Page 760 Overall Solutions......Page 762 Laplace Expansion......Page 764 Symmetry of Solutions......Page 766 Further Properties......Page 768 15.6 Legendre Functions of the Second Kind......Page 770 Alternate Formulations......Page 773 Additional Readings......Page 775 16 Angular Momentum......Page 777 16.1 Angular Momentum Operators......Page 778 Ladder Operators......Page 780 Spinors......Page 783 Summary, Angular Momentum Formulas......Page 785 16.2 Angular Momentum Coupling......Page 788 Vector Model......Page 790 Ladder Operator Construction......Page 792 16.3 Spherical Tensors......Page 800 Addition Theorem......Page 801 Spherical Wave Expansion......Page 802 Laplace Spherical Harmonic Expansion......Page 803 General Multipoles......Page 805 Integrals of Three Spherical Harmonics......Page 807 A Spherical Tensor......Page 813 Vector Coupling......Page 814 Additional Readings......Page 818 17.1 Introduction to Group Theory......Page 819 Definition of a Group......Page 820 Examples of Groups......Page 821 17.2 Representation of Groups......Page 825 17.3 Symmetry and Physics......Page 830 Classes......Page 834 Other Discrete Groups......Page 839 17.5 Direct Products......Page 841 17.6 Symmetric Group......Page 844 17.7 Continuous Groups......Page 849 Lie Groups and Their Generators......Page 850 Groups SO(2) and SO(3)......Page 853 Group SU(2) and SU(2)–SO(3) Homomorphism......Page 855 Group SU(3)......Page 856 Homogeneous Lorentz Group......Page 866 Minkowski Space......Page 868 17.9 Lorentz Covariance of Maxwell's Equations......Page 870 Lorentz Transformation of E and B......Page 871 17.10 Space Groups......Page 873 Additional Readings......Page 874 18.1 Hermite Functions......Page 875 Recurrence Relations......Page 876 Special Values......Page 877 Rodrigues Formula......Page 878 Orthogonality and Normalization......Page 879 Simple Harmonic Oscillator......Page 882 Operator Approach......Page 883 Molecular Vibrations......Page 886 Hermite Product Formula......Page 888 Rodrigues Formula and Generating Function......Page 893 Properties of Laguerre Polynomials......Page 894 Associated Laguerre Polynomials......Page 896 Type II Polynomials......Page 903 Type I Polynomials......Page 904 Recurrence Relations......Page 905 Special Values......Page 907 Trigonometric Form......Page 908 Application to Numerical Analysis......Page 909 Orthogonality......Page 910 18.5 Hypergeometric Functions......Page 915 Hypergeometric Representations......Page 917 18.6 Confluent Hypergeometric Functions......Page 921 Confluent Hypergeometric Representations......Page 922 Further Observations......Page 923 Expansion and Analytic Properties......Page 927 Properties and Special Values......Page 928 18.8 Elliptic Integrals......Page 931 Definitions......Page 932 Series Expansions......Page 933 Limiting Values......Page 934 Additional Readings......Page 936 19.1 General Properties......Page 938 Sturm-Liouville Theory......Page 939 Discontinuous Functions......Page 940 Symmetry......Page 943 Operations on Fourier Series......Page 945 Summing Fourier Series......Page 947 19.2 Applications of Fourier Series......Page 952 Partial Summation of Fourier Series......Page 960 Square Wave......Page 961 Calculation of Overshoot......Page 962 Additional Readings......Page 965 20.1 Introduction......Page 966 Some Important Transforms......Page 968 20.2 Fourier Transform......Page 969 Fourier Integral......Page 972 Inverse Fourier Transform......Page 973 Transforms in 3-D Space......Page 976 20.3 Properties of Fourier Transforms......Page 983 Successes and Limitations......Page 987 20.4 Fourier Convolution Theorem......Page 988 Parseval Relation......Page 990 Multiple Convolutions......Page 993 Transform of a Product......Page 995 Momentum Space......Page 996 20.5 Signal-Processing Applications......Page 1000 Limitations on Transfer Functions......Page 1003 20.6 Discrete Fourier Transform......Page 1005 Orthogonality on Discrete Point Sets......Page 1006 Discrete Fourier Transform......Page 1007 Limitations......Page 1008 Fast Fourier Transform......Page 1009 Elementary Functions......Page 1011 Dirac Delta Function......Page 1013 Inverse Transform......Page 1014 Transforms of Derivatives......Page 1019 Substitution......Page 1023 Translation......Page 1025 Derivative of a Transform......Page 1027 Integration of Transforms......Page 1030 20.9 Laplace Convolution Theorem......Page 1037 Bromwich Integral......Page 1041 Additional Readings......Page 1048 21.1 Introduction......Page 1050 Transformation of a Differential Equation into an Integral Equation......Page 1052 21.2 Some Special Methods......Page 1056 Integral-Transform Methods......Page 1057 Generating-Function Method......Page 1059 Separable Kernel......Page 1060 21.3 Neumann Series......Page 1067 Orthogonal Eigenfunctions......Page 1072 Inhomogeneous Integral Equation......Page 1076 Additional Readings......Page 1082 22.1 Euler Equation......Page 1083 Alternate Forms of Euler Equations......Page 1090 Soap Film: Minimum Area......Page 1092 Several Dependent Variables......Page 1098 Hamilton's Principle......Page 1099 Hamilton's Equations......Page 1101 Several Independent Variables......Page 1102 Several Dependent and Independent Variables......Page 1104 Geodesics......Page 1105 Relation to Physics......Page 1107 22.3 Constrained Minima/Maxima......Page 1109 Lagrangian Multipliers......Page 1110 22.4 Variation with Constraints......Page 1113 Lagrangian Formulation with Constraints......Page 1114 Rayleigh-Ritz Technique......Page 1119 Ground State Eigenfunction......Page 1120 Additional Readings......Page 1126 23 Probability and Statistics......Page 1127 23.1 Probability: Definitions, Simple Properties......Page 1128 Sets, Unions, and Intersections......Page 1129 Counting Permutations and Combinations......Page 1132 23.2 Random Variables......Page 1136 Mean and Variance......Page 1138 Moments of Probability Distributions......Page 1143 Covariance and Correlation......Page 1144 Marginal Probability Distributions......Page 1146 Conditional Probability Distributions......Page 1149 23.3 Binomial Distribution......Page 1150 23.4 Poisson Distribution......Page 1153 Relation to Binomial Distribution......Page 1155 23.5 Gauss' Normal Distribution......Page 1157 Limits of Poisson and Binomial Distributions......Page 1159 23.6 Transformations of Random Variables......Page 1161 Addition of Random Variables......Page 1162 Gamma Distribution......Page 1164 Error Propagation......Page 1167 Fitting Curves to Data......Page 1170 The x2 Distribution......Page 1172 Student t Distribution......Page 1176 Confidence Intervals......Page 1178 Additional Readings......Page 1181 A......Page 1182 B......Page 1183 C......Page 1184 D......Page 1187 E......Page 1188 F......Page 1189 G......Page 1190 H......Page 1192 I......Page 1193 L......Page 1195 M......Page 1197 O......Page 1198 P......Page 1199 R......Page 1201 S......Page 1202 T......Page 1204 V......Page 1205 Z......Page 1206 Page 1......Page 1 Providing coverage of the mathematics necessary for advanced study in physics and engineering, this text focuses on problem-solving skills and offers a vast array of exercises, as well as clearly illustrating and proving mathematical relations. Now inits 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining thekey features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises. Revised and updated version of the leading text in mathematical physicsFocuses on problem-solving skills and active learning, offering numerous chapter problemsClearly identified definitions, theorems, and proofs promote clarity and understanding New to this edition: Improved modular chaptersNew up-to-date examplesMore intuitive explanations

Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises.

• Revised and updated version of the leading text in mathematical physics
• Focuses on problem-solving skills and active learning, offering numerous chapter problems
• Clearly identified definitions, theorems, and proofs promote clarity and understanding

New to this edition:

• Improved modular chapters
• New up-to-date examples
• More intuitive explanations

Mathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on the mathematics of theoretical physics.
This edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations, numerical application of Chebyshev polynomials, the fast Fourier transform, and transfer functions. Many of the physical examples provided in this book, which are used to illustrate the applications of mathematics, are taken from the fields of electromagnetic theory and quantum mechanics. The Hermitian operators, Hilbert space, and concept of completeness are also deliberated.
This book is beneficial to students studying graduate level physics, particularly theoretical physics.