Mathematical Methods for Physics and Engineering: A Comprehensive Guide (2nd Edition)

Half-title......Page 3
Dedication......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the second edition......Page 21
Preface to the first edition......Page 23
1.1.1 Polynomials and polynomial equations......Page 27
An nth-degree polynomial equation has exactly n roots.......Page 30
A more general case......Page 32
1.1.2 Factorising polynomials......Page 33
1.1.3 Properties of roots......Page 35
1.2.1 Single-angle identities......Page 36
1.2.2 Compound-angle identities......Page 37
1.2.3 Double-and half-angle identities......Page 39
1.3 Coordinate geometry......Page 41
1.4 Partial fractions......Page 44
The degree of the numerator is greater than or equal to that of the denominator......Page 47
Factors of the form a2 + x2 in the denominator......Page 48
Repeated factors in the denominator......Page 49
1.5 Binomial expansion......Page 51
1.5.2 Proof of the binomial expansion......Page 52
1.6.1 Identities involving binomial coefficients......Page 53
1.6.2 Negative and non-integral values of n......Page 55
1.7 Some particular methods of proof......Page 56
1.7.1 Proof by induction......Page 57
1.7.2 Proof by contradiction......Page 58
1.7.3 Necessary and sufficient conditions......Page 60
Trigonometric identities......Page 62
Partial fractions......Page 63
Proof by induction and contradiction......Page 64
1.9 Hints and answers......Page 65
2.1.1 Differentiation from first principles......Page 68
2.1.2 Differentiation of products......Page 71
2.1.3 The chain rule......Page 73
2.1.5 Implicit differentiation......Page 74
2.1.7 Leibnitz’ theorem......Page 75
2.1.8 Special points of a function......Page 77
2.1.9 Curvature of a function......Page 79
Rolle’s theorem......Page 82
Mean value theorem......Page 83
Applications of Rolle’s theorem and the mean value theorem......Page 84
2.2.1 Integration from first principles......Page 86
2.2.2 Integration as the inverse of differentiation......Page 88
2.2.3 Integration by inspection......Page 89
2.2.4 Integration of sinusoidal functions......Page 90
2.2.6 Integration using partial fractions......Page 91
2.2.7 Integration by substitution......Page 92
2.2.8 Integration by parts......Page 94
2.2.9 Reduction formulae......Page 96
2.2.11 Integration in plane polar coordinates......Page 97
Mean value of a function......Page 99
Finding the length of a curve......Page 100
Surfaces of revolution......Page 101
Volumes of revolution......Page 102
2.3 Exercises......Page 103
2.4 Hints and answers......Page 108
3.1 The need for complex numbers......Page 112
3.2.1 Addition and subtraction......Page 114
3.2.2 Modulus and argument......Page 116
3.2.3 Multiplication......Page 117
3.2.4 Complex conjugate......Page 118
3.2.5 Division......Page 120
3.3 Polar representation of complex numbers......Page 121
3.3.1 Multiplication and division in polar form......Page 123
3.4.1 Trigonometric identities......Page 124
3.4.2 Finding the nth roots of unity......Page 126
3.4.3 Solving polynomial equations......Page 127
3.5 Complex logarithms and complex powers......Page 128
3.6 Applications to differentiation and integration......Page 130
3.7.2 Hyperbolic-trigonometric analogies......Page 131
3.7.3 Identities of hyperbolic functions......Page 133
3.7.5 Inverses of hyperbolic functions......Page 134
3.7.6 Calculus of hyperbolic functions......Page 135
3.8 Exercises......Page 138
3.9 Hints and answers......Page 142
4.1 Series......Page 144
4.2 Summation of series......Page 145
4.2.2 Geometric series......Page 146
4.2.3 Arithmetico-geometric series......Page 147
4.2.4 The difference method......Page 148
4.2.5 Series involving natural numbers......Page 150
4.2.6 Transformation of series......Page 151
4.3.1 Absolute and conditional convergence......Page 153
Comparison test......Page 154
D’Alembert’s ratio test......Page 155
Quotient test......Page 156
Integral test......Page 157
Grouping terms......Page 158
4.3.3 Alternating series test......Page 159
4.5 Power series......Page 160
4.5.1 Convergence of power series......Page 161
4.5.2 Operations with power series......Page 163
4.6.1 Taylor’s theorem......Page 165
4.6.2 Approximation errors in Taylor series......Page 168
4.6.3 Standard Maclaurin series......Page 169
4.7 Evaluation of limits......Page 170
4.8 Exercises......Page 173
4.9 Hints and answers......Page 178
5.1 Definition of the partial derivative......Page 180
5.2 The total differential and total derivative......Page 182
5.3 Exact and inexact differentials......Page 184
5.5 The chain rule......Page 186
5.6 Change of variables......Page 187
5.7 Taylor’s theorem for many-variable functions......Page 189
5.8 Stationary values of many-variable functions......Page 191
5.9 Stationary values under constraints......Page 196
5.10 Envelopes......Page 202
5.10.1 Envelope equations......Page 203
5.11 Thermodynamic relations......Page 205
5.12 Differentiation of integrals......Page 207
5.13 Exercises......Page 208
5.14 Hints and answers......Page 214
6.1 Double integrals......Page 216
6.2 Triple integrals......Page 219
6.3.1 Areas and volumes......Page 220
6.3.2 Masses, centres of mass and centroids......Page 222
6.3.3 Pappus’ theorems......Page 224
6.3.4 Moments of inertia......Page 227
6.4 Change of variables in multiple integrals......Page 228
6.4.1 Change of variables in double integrals......Page 229
6.4.2 Evaluation of the integral…......Page 231
6.4.3 Change of variables in triple integrals......Page 233
6.4.4 General properties of Jacobians......Page 235
6.5 Exercises......Page 236
6.6 Hints and answers......Page 240
7.1 Scalars and vectors......Page 242
7.2 Addition and subtraction of vectors......Page 243
7.3 Multiplication by a scalar......Page 244
7.4 Basis vectors and components......Page 247
7.5 Magnitude of a vector......Page 248
7.6.1 Scalar product......Page 249
7.6.2 Vector product......Page 252
7.6.3 Scalar triple product......Page 254
7.7.1 Equation of a line......Page 256
7.7.2 Equation of a plane......Page 257
7.7.3 Equation of a sphere......Page 258
7.8.1 Distance from a point to a line......Page 259
7.8.2 Distance from a point to a plane......Page 260
7.8.3 Distance from a line to a line......Page 261
7.8.4 Distance from a line to a plane......Page 262
7.9 Reciprocal vectors......Page 263
7.10 Exercises......Page 264
7.11 Hints and answers......Page 270
8 Matrices and vector spaces......Page 272
8.1 Vector spaces......Page 273
8.1.1 Basis vectors......Page 274
8.1.2 The inner product......Page 275
8.1.3 Some useful inequalities......Page 277
8.2 Linear operators......Page 278
8.3 Matrices......Page 280
8.4 Basic matrix algebra......Page 281
8.4.1 Matrix addition and multiplication by a scalar......Page 282
8.4.2 Multiplication of matrices......Page 283
8.4.3 The null and identity matrices......Page 285
8.6 The transpose of a matrix......Page 286
8.7 The complex and Hermitian conjugates of a matrix......Page 287
8.8 The trace of a matrix......Page 289
8.9 The determinant of a matrix......Page 290
8.9.1 Properties of determinants......Page 292
8.10 The inverse of a matrix......Page 294
8.11 The rank of a matrix......Page 298
8.12.1 Diagonal matrices......Page 299
8.12.2 Lower and upper triangular matrices......Page 300
8.12.4 Orthogonal matrices......Page 301
8.12.6 Unitary matrices......Page 302
8.13 Eigenvectors and eigenvalues......Page 303
8.13.1 Eigenvectors and eigenvalues of a normal matrix......Page 304
8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices......Page 307
8.13.5 Simultaneous eigenvectors......Page 309
8.14 Determination of eigenvalues and eigenvectors......Page 311
8.14.1 Degenerate eigenvalues......Page 313
8.15 Change of basis and similarity transformations......Page 314
8.16 Diagonalisation of matrices......Page 316
8.17 Quadratic and Hermitian forms......Page 319
8.17.1 The stationary properties of the eigenvectors......Page 321
8.18 Simultaneous linear equations......Page 323
8.18.1 The range and null space of a matrix......Page 324
Infinitely many solutions......Page 325
Direct inversion......Page 326
LU decomposition......Page 327
Cramer’s rule......Page 330
8.18.3 Singular value decomposition......Page 332
8.19 Exercises......Page 338
8.20 Hints and answers......Page 345
9 Normal modes......Page 348
9.1 Typical oscillatory systems......Page 349
9.2 Symmetry and normal modes......Page 354
9.3 Rayleigh–Ritz method......Page 359
9.4 Exercises......Page 361
9.5 Hints and answers......Page 364
10.1 Differentiation of vectors......Page 366
10.1.1 Differentiation of composite vector expressions......Page 369
10.1.2 Differential of a vector......Page 370
10.2 Integration of vectors......Page 371
10.3 Space curves......Page 372
10.4 Vector functions of several arguments......Page 376
10.5 Surfaces......Page 377
10.7 Vector operators......Page 379
10.7.1 Gradient of a scalar field......Page 380
10.7.2 Divergence of a vector field......Page 384
10.7.3 Curl of a vector field......Page 385
10.8.1 Vector operators acting on sums and products......Page 386
10.8.2 Combinations of grad, div and curl......Page 387
10.9.1 Cylindrical polar coordinates......Page 389
10.9.2 Spherical polar coordinates......Page 393
10.10 General curvilinear coordinates......Page 396
Divergence......Page 399
Curl......Page 400
10.11 Exercises......Page 401
10.12 Hints and answers......Page 407
11.1 Line integrals......Page 409
11.1.1 Evaluating line integrals......Page 410
11.1.2 Physical examples of line integrals......Page 413
11.1.3 Line integrals with respect to a scalar......Page 414
11.2 Connectivity of regions......Page 415
11.3 Green’s theorem in a plane......Page 416
11.4 Conservative fields and potentials......Page 419
11.5 Surface integrals......Page 421
11.5.1 Evaluating surface integrals......Page 423
11.5.2 Vector areas of surfaces......Page 425
11.5.3 Physical examples of surface integrals......Page 427
11.6 Volume integrals......Page 428
11.6.1 Volumes of three-dimensional regions......Page 429
11.7 Integral forms for grad, div and curl......Page 430
11.8 Divergence theorem and related theorems......Page 433
11.8.1 Green’s theorems......Page 434
11.8.2 Other related integral theorems......Page 435
11.8.3 Physical applications of the divergence theorem......Page 436
11.9 Stokes’ theorem and related theorems......Page 438
11.9.1 Related integral theorems......Page 439
11.9.2 Physical applications of Stokes’ theorem......Page 440
11.10 Exercises......Page 441
11.11 Hints and answers......Page 446
12.1 The Dirichlet conditions......Page 447
12.2 The Fourier coefficients......Page 449
12.3 Symmetry considerations......Page 451
12.4 Discontinuous functions......Page 452
12.5 Non-periodic functions......Page 454
12.7 Complex Fourier series......Page 456
12.8 Parseval’s theorem......Page 458
12.9 Exercises......Page 459
12.10 Hints and answers......Page 463
13.1 Fourier transforms......Page 465
13.1.1 The uncertainty principle......Page 467
13.1.2 Fraunhofer diffraction......Page 469
13.1.3 The Dirac Delta-function......Page 471
13.1.4 Relation of the Delta-function to Fourier transforms......Page 474
13.1.5 Properties of Fourier transforms......Page 475
13.1.6 Odd and even functions......Page 477
13.1.7 Convolution and deconvolution......Page 478
13.1.8 Correlation functions and energy spectra......Page 481
13.1.9 Parseval’s theorem......Page 482
13.1.10 Fourier transforms in higher dimensions......Page 483
13.2 Laplace transforms......Page 485
13.2.1 Laplace transforms of derivatives and integrals......Page 487
13.2.2 Other properties of Laplace transforms......Page 488
13.3 Concluding remarks......Page 491
13.4 Exercises......Page 492
13.5 Hints and answers......Page 498
14 First-order ordinary differential equations......Page 500
14.1 General form of solution......Page 501
14.2 First-degree first-order equations......Page 502
14.2.1 Separable-variable equations......Page 503
14.2.2 Exact equations......Page 504
14.2.3 Inexact equations: integrating factors......Page 505
14.2.4 Linear equations......Page 506
14.2.5 Homogeneous equations......Page 507
14.2.6 Isobaric equations......Page 508
14.2.7 Bernoulli’s equation......Page 509
14.2.8 Miscellaneous equations......Page 510
14.3.1 Equations soluble for p......Page 512
14.3.2 Equations soluble for x......Page 513
14.3.3 Equations soluble for y......Page 514
14.3.4 Clairaut’s equation......Page 515
14.4 Exercises......Page 516
14.5 Hints and answers......Page 520
15 Higher-order ordinary differential equations......Page 522
15.1.1 Finding the complementary function yc(x)......Page 524
15.1.2 Finding the particular integral yp(x)......Page 526
15.1.3 Constructing the general solution yc(x) + yp(x)......Page 527
15.1.4 Linear recurrence relations......Page 528
First-order recurrence relations......Page 529
Second-order recurrence relations......Page 531
15.1.5 Laplace transform method......Page 533
15.2.1 The Legendre and Euler linear equations......Page 535
15.2.2 Exact equations......Page 537
15.2.3 Partially known complementary function......Page 538
15.2.4 Variation of parameters......Page 540
15.2.5 Green’s functions......Page 543
15.2.6 Canonical form for second-order equations......Page 548
15.3.2 Independent variable absent......Page 550
15.3.3 Non-linear exact equations......Page 551
15.3.4 Isobaric or homogeneous equations......Page 553
15.3.5 Equations homogeneous in x or y alone......Page 554
15.4 Exercises......Page 555
15.5 Hints and answers......Page 561
16.1 Second-order linear ordinary differential equations......Page 563
16.1.1 Ordinary and singular points of an ODE......Page 565
16.2 Series solutions about an ordinary point......Page 567
16.3 Series solutions about a regular singular point......Page 570
16.3.1 Distinct roots not differing by an integer......Page 572
16.3.2 Repeated root of the indicial equation......Page 573
16.3.3 Distinct roots differing by an integer......Page 574
16.4 Obtaining a second solution......Page 575
16.4.1 The Wronskian method......Page 576
16.4.2 The derivative method......Page 577
16.4.3 Series form of the second solution......Page 579
16.5 Polynomial solutions......Page 580
16.6 Legendre’s equation......Page 581
16.6.1 General solution for integer l......Page 582
Rodrigues’ formula......Page 585
Mutual orthogonality of Legendre polynomials......Page 586
Generating function for Legendre polynomials......Page 588
16.7 Bessel’s equation......Page 590
16.7.1 General solution for non-integer v......Page 591
16.7.2 General solution for integer v......Page 593
16.7.3 Properties of Bessel functions......Page 594
Recurrence relations......Page 595
Mutual orthogonality of Bessel functions......Page 596
Generating function for Bessel functions......Page 599
16.9 Exercises......Page 601
16.10 Hints and answers......Page 605
17 Eigenfunction methods for differential equations......Page 607
17.1 Sets of functions......Page 609
17.1.1 Some useful inequalities......Page 612
17.2 Adjoint and Hermitian operators......Page 613
17.3.1 Reality of the eigenvalues......Page 614
17.3.2 Orthogonality of the eigenfunctions......Page 615
17.3.3 Construction of real eigenfunctions......Page 616
17.4 Sturm–Liouville equations......Page 617
17.4.2 Putting an equation into Sturm–Liouville form......Page 618
17.5.1 Legendre’s equation......Page 619
17.5.2 The associated Legendre equation......Page 620
17.5.4 The simple harmonic equation......Page 621
17.5.6 Laguerre’s equation......Page 622
17.6 Superposition of eigenfunctions: Green’s functions......Page 623
17.7 A useful generalisation......Page 627
17.8 Exercises......Page 628
17.9 Hints and answers......Page 632
18 Partial differential equations: general and particular solutions......Page 634
18.1.1 The wave equation......Page 635
18.1.2 The diffusion equation......Page 637
18.1.5 Schrödinger’s equation......Page 638
18.2 General form of solution......Page 639
18.3.1 First-order equations......Page 640
18.3.2 Inhomogeneous equations and problems......Page 644
18.3.3 Second-order equations......Page 646
18.4 The wave equation......Page 652
18.5 The diffusion equation......Page 654
18.6.1 First-order equations......Page 658
18.6.2 Second-order equations......Page 660
18.7 Uniqueness of solutions......Page 664
18.8 Exercises......Page 666
18.9 Hints and answers......Page 670
19.1 Separation of variables: the general method......Page 672
19.2 Superposition of separated solutions......Page 676
Laplace’s equation in plane polars......Page 684
Laplace’s equation in cylindrical polars......Page 687
Laplace’s equation in spherical polars......Page 690
19.3.2 Spherical harmonics......Page 696
19.3.3 Other equations in polar coordinates......Page 697
Helmholtz’s equation in plane polars......Page 698
Helmholtz’s equation in cylindrical polars......Page 699
Helmholtz’s equation in spherical polars......Page 700
19.3.4 Solution by expansion......Page 702
19.3.5 Separation of variables for inhomogeneous equations......Page 704
19.4 Integral transform methods......Page 707
19.5 Inhomogeneous problems – Green’s functions......Page 712
19.5.1 Similarities to Green’s functions for ODEs......Page 713
19.5.2 General boundary-value problems......Page 714
19.5.3 Dirichlet problems......Page 716
19.5.4 Neumann problems......Page 726
19.6 Exercises......Page 728
19.7 Hints and answers......Page 734
20 Complex variables......Page 736
20.1 Functions of a complex variable......Page 737
20.2 The Cauchy–Riemann relations......Page 739
20.3 Power series in a complex variable......Page 742
20.4 Some elementary functions......Page 744
20.5 Multivalued functions and branch cuts......Page 747
20.6 Singularities and zeroes of complex functions......Page 749
20.7 Complex potentials......Page 751
20.8 Conformal transformations......Page 756
20.9 Applications of conformal transformations......Page 761
20.10 Complex integrals......Page 764
20.11 Cauchy’s theorem......Page 768
20.12 Cauchy’s integral formula......Page 771
20.13 Taylor and Laurent series......Page 773
20.14 Residue theorem......Page 778
20.15 Location of zeroes......Page 780
20.16 Integrals of sinusoidal functions......Page 784
20.17 Some infinite integrals......Page 785
20.18 Integrals of multivalued functions......Page 788
20.19 Summation of series......Page 790
20.20 Inverse Laplace transform......Page 791
20.21 Exercises......Page 794
20.22 Hints and answers......Page 799
21 Tensors......Page 802
21.1 Some notation......Page 803
21.2 Change of basis......Page 804
21.3 Cartesian tensors......Page 805
21.4 First-and zero-order Cartesian tensors......Page 807
21.5 Second- and higher-order Cartesian tensors......Page 810
21.6 The algebra of tensors......Page 813
21.7 The quotient law......Page 814
21.8 The tensors…......Page 816
21.9 Isotropic tensors......Page 819
21.10 Improper rotations and pseudotensors......Page 821
21.11 Dual tensors......Page 824
21.12 Physical applications of tensors......Page 825
21.13 Integral theorems for tensors......Page 829
21.14 Non-Cartesian coordinates......Page 830
21.15 The metric tensor......Page 832
21.16 General coordinate transformations and tensors......Page 835
21.17 Relative tensors......Page 838
21.18 Derivatives of basis vectors and Christoffel symbols......Page 840
21.19 Covariant differentiation......Page 843
21.20 Vector operators in tensor form......Page 846
Divergence......Page 847
Laplacian......Page 848
Curl......Page 849
21.21 Absolute derivatives along curves......Page 850
21.22 Geodesics......Page 851
21.23 Exercises......Page 852
21.24 Hints and answers......Page 857
22 Calculus of variations......Page 860
22.1 The Euler–Lagrange equation......Page 861
22.2.1 F does not contain y explicitly......Page 862
22.2.2 F does not contain x explicitly......Page 864
22.3 Some extensions......Page 866
22.3.4 Variable end-points......Page 867
22.4 Constrained variation......Page 870
22.5.1 Fermat’s principle in optics......Page 872
22.5.2 Hamilton’s principle in mechanics......Page 873
22.6 General eigenvalue problems......Page 875
22.7 Estimation of eigenvalues and eigenfunctions......Page 877
22.8 Adjustment of parameters......Page 880
22.9 Exercises......Page 882
22.10 Hints and answers......Page 886
23.1 Obtaining an integral equation from a differential equation......Page 888
23.2 Types of integral equation......Page 889
23.3 Operator notation and the existence of solutions......Page 890
23.4 Closed-form solutions......Page 891
23.4.1 Separable kernels......Page 892
23.4.2 Integral transform methods......Page 894
23.4.3 Differentiation......Page 897
23.5 Neumann series......Page 898
23.6 Fredholm theory......Page 900
23.7 Schmidt–Hilbert theory......Page 901
23.8 Exercises......Page 904
23.9 Hints and answers......Page 908
24.1 Groups......Page 909
24.1.1 Definition of a group......Page 911
24.1.2 Further examples of groups......Page 916
24.2 Finite groups......Page 917
24.3 Non-Abelian groups......Page 920
24.4 Permutation groups......Page 924
24.5 Mappings between groups......Page 927
24.6 Subgroups......Page 929
24.7 Subdividing a group......Page 931
24.7.1 Equivalence relations and classes......Page 932
24.7.2 Congruence and cosets......Page 933
24.7.3 Conjugates and classes......Page 936
24.8 Exercises......Page 938
24.9 Hints and answers......Page 941
25 Representation theory......Page 944
25.1 Dipole moments of molecules......Page 945
25.2 Choosing an appropriate formalism......Page 946
25.3 Equivalent representations......Page 952
25.4 Reducibility of a representation......Page 954
25.5 The orthogonality theorem for irreducible representations......Page 958
25.6 Characters......Page 960
25.6.1 Orthogonality property of characters......Page 962
25.7 Counting irreps using characters......Page 963
25.7.1 Summation rules for irreps......Page 965
25.8 Construction of a character table......Page 968
25.9 Group nomenclature......Page 970
25.10 Product representations......Page 971
25.11 Physical applications of group theory......Page 973
25.11.1 Bonding in molecules......Page 974
25.11.2 Matrix elements in quantum mechanics......Page 976
25.11.3 Degeneracy of normal modes......Page 978
25.11.4 Breaking of degeneracies......Page 979
25.12 Exercises......Page 981
25.13 Hints and answers......Page 985
26.1 Venn diagrams......Page 987
26.2 Probability......Page 992
26.2.1 Axioms and theorems......Page 993
26.2.2 Conditional probability......Page 996
26.2.3 Bayes’ theorem......Page 1000
26.3.1 Permutations......Page 1001
26.3.2 Combinations......Page 1003
26.4.1 Discrete random variables......Page 1007
26.4.2 Continuous random variables......Page 1008
26.4.3 Sets of random variables......Page 1010
26.5 Properties of distributions......Page 1011
26.5.1 Mean......Page 1012
26.5.2 Mode and median......Page 1013
26.5.3 Variance and standard deviation......Page 1014
26.5.4 Moments......Page 1015
26.5.5 Central moments......Page 1016
26.6 Functions of random variables......Page 1018
26.6.2 Continuous random variables......Page 1019
26.6.3 Functions of several random variables......Page 1021
26.6.4 Expectation values and variances......Page 1023
26.7.1 Probability generating functions......Page 1025
Sums of random variables......Page 1028
Variable-length sums of random variables......Page 1029
26.7.2 Moment generating functions......Page 1030
Variable-length sums of random variables......Page 1032
26.7.3 Characteristic function......Page 1033
26.7.4 Cumulant generating function......Page 1034
26.8 Important discrete distributions......Page 1035
26.8.1 The binomial distribution......Page 1036
The moment generating function for the binomial distribution......Page 1038
Multiple binomial distributions......Page 1039
26.8.2 The geometric and negative binomial distributions......Page 1040
26.8.3 The hypergeometric distribution......Page 1041
26.8.4 The Poisson distribution......Page 1042
The Poisson approximation to the binomial distribution......Page 1045
Multiple Poisson distributions......Page 1046
26.9.1 The Gaussian distribution......Page 1047
The moment generating function for the Gaussian distribution......Page 1052
Gaussian approximation to the binomial distribution......Page 1053
Gaussian approximation to the Poisson distribution......Page 1055
Multiple Gaussian distributions......Page 1056
26.9.2 The log-normal distribution......Page 1057
26.9.3 The exponential and gamma distributions......Page 1058
26.9.4 The chi-squared distribution......Page 1060
26.9.5 The Cauchy and Breit–Wigner distributions......Page 1061
26.10 The central limit theorem......Page 1062
26.11 Joint distributions......Page 1064
26.11.2 Continuous bivariate distributions......Page 1065
26.11.3 Marginal and conditional distributions......Page 1066
26.12.1 Means......Page 1067
26.12.3 Covariance and correlation......Page 1068
26.13 Generating functions for joint distributions......Page 1073
26.14 Transformation of variables in joint distributions......Page 1074
26.15 Important joint distributions......Page 1075
26.15.1 The multinomial distribution......Page 1076
26.15.2 The multivariate Gaussian distribution......Page 1077
26.16 Exercises......Page 1079
26.17 Hints and answers......Page 1087
27.1 Experiments, samples and populations......Page 1090
27.2.1 Averages......Page 1091
27.2.2 Variance and standard deviation......Page 1093
27.2.3 Moments and central moments......Page 1095
27.2.4 Covariance and correlation......Page 1096
27.3 Estimators and sampling distributions......Page 1098
Consistency......Page 1099
Efficiency......Page 1100
27.3.2 Fisher’s inequality......Page 1102
27.3.3 Standard errors on estimators......Page 1103
27.3.4 Confidence limits on estimators......Page 1104
27.3.5 Confidence limits for a Gaussian sampling distribution......Page 1106
27.3.6 Estimation of several quantities simultaneously......Page 1108
27.4.1 Population mean Mu......Page 1112
27.4.2 Population variance Sigma2......Page 1113
27.4.3 Population standard deviation Sigma......Page 1116
27.4.4 Population moments Mur......Page 1117
27.4.5 Population central moments vr......Page 1118
27.4.6 Population covariance Cov[x, y] and correlation Corr[x, y]......Page 1119
27.4.7 A worked example......Page 1121
27.5 Maximum-likelihood method......Page 1123
27.5.1 The maximum-likelihood estimator......Page 1124
27.5.2 Transformation invariance and bias of ML estimators......Page 1128
27.5.3 Efficiency of ML estimators......Page 1129
27.5.4 Standard errors and confidence limits on ML estimators......Page 1130
27.5.5 The Bayesian interpretation of the likelihood function......Page 1132
27.5.6 Behaviour of ML estimators for large N......Page 1137
27.5.7 Extended maximum-likelihood method......Page 1138
27.6 The method of least squares......Page 1139
27.6.1 Linear least squares......Page 1140
27.6.2 Non-linear least squares......Page 1144
27.7 Hypothesis testing......Page 1145
27.7.2 Statistical tests......Page 1146
27.7.3 The Neyman–Pearson test......Page 1148
27.7.4 The generalised likelihood-ratio test......Page 1149
27.7.5 Student’s t-test......Page 1152
27.7.6 Fisher’s F-test......Page 1158
27.7.7 Goodness of fit in least-squares problems......Page 1164
27.8 Exercises......Page 1166
27.9 Hints and answers......Page 1171
28 Numerical methods......Page 1174
28.1 Algebraic and transcendental equations......Page 1175
28.1.1 Rearrangement of the equation......Page 1177
28.1.2 Linear interpolation......Page 1178
28.1.4 Newton–Raphson method......Page 1180
28.2 Convergence of iteration schemes......Page 1182
28.3 Simultaneous linear equations......Page 1184
28.3.1 Gaussian elimination......Page 1185
28.3.2 Gauss–Seidel iteration......Page 1186
28.3.3 Tridiagonal matrices......Page 1188
28.4 Numerical integration......Page 1190
28.4.1 Trapezium rule......Page 1192
28.4.2 Simpson’s rule......Page 1193
28.4.3 Gaussian integration......Page 1194
28.4.4 Monte Carlo methods......Page 1196
Importance sampling......Page 1198
Control variates......Page 1199
Hit or miss method......Page 1200
Random number generation......Page 1203
28.5 Finite differences......Page 1205
28.6 Differential equations......Page 1206
28.6.1 Difference equations......Page 1207
28.6.2 Taylor series solutions......Page 1209
28.6.3 Prediction and correction......Page 1210
28.6.4 Runge–Kutta methods......Page 1212
28.7 Higher-order equations......Page 1214
28.8 Partial differential equations......Page 1216
28.9 Exercises......Page 1219
28.10 Hints and answers......Page 1224
A1.1 The gamma function......Page 1227
A1.2 The beta function......Page 1229
A1.3 The error function......Page 1230
Index......Page 1232