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"
Author(s): George B. Arfken, Hans J. Weber, Frank E. Harris
Edition: Seventh Edition
Publisher: Academic Press
Year: 2012
Language: English
Pages: 1205
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
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