Mathematics for Physics and Physicists

Contents......Page 6
Thanks......Page 18
A book’s apology......Page 19
Notation......Page 23
1.1.a Two paradoxes involving kinetic energy......Page 26
1.1.b Romeo, Juliet, and viscous fluids......Page 30
1.1.c Potential wall in quantum mechanics......Page 32
1.1.d Semi-infinite filter behaving as waveguide......Page 34
1.2.a Sequences in a normed vector space......Page 37
1.2.b Cauchy sequences......Page 38
1.2.c The fixed point theorem......Page 40
1.2.d Double sequences......Page 41
1.2.e Sequential definition of the limit of a function......Page 42
1.2.f Sequences of functions......Page 43
1.3.a Series in a normed vector space......Page 48
1.3.b Doubly infinite series......Page 49
1.3.c Convergence of a double series......Page 50
1.3.d Conditionally convergent series, absolutely convergent series......Page 51
1.3.e Series of functions......Page 54
1.4 Power series, analytic functions......Page 55
1.4.a Taylor formulas......Page 56
1.4.b Some numerical illustrations......Page 57
1.4.c Radius of convergence of a power series......Page 59
1.4.d Analytic functions......Page 60
1.5.a Asymptotic series......Page 62
1.5.b Divergent series and asymptotic expansions......Page 63
2.1.a Riemann sums......Page 76
2.2 The integral according to Mr. Lebesgue......Page 79
2.2.a Principle of the method......Page 80
2.2.b Borel subsets......Page 82
2.2.c Lebesgue measure......Page 84
2.2.d The Lebesgue σ-algebra......Page 85
2.2.e Negligible sets......Page 86
2.2.f Lebesgue measure on R......Page 87
2.2.g Definition of the Lebesgue integral......Page 88
2.2.h Functions zero almost everywhere, space L^1 of integrable functions......Page 91
2.2.i And today?......Page 92
3.1.a Standard functions......Page 98
3.1.b Comparison theorems......Page 99
3.2 Exchanging integrals and limits or series......Page 100
3.3.a Continuity of functions defined by integrals......Page 102
3.3.c Case of parameters appearing in the integration range......Page 103
3.4 Double and multiple integrals......Page 104
3.5 Change of variables......Page 106
4.1 Holomorphic functions......Page 112
4.1.a Definitions......Page 113
4.1.b Examples......Page 115
4.1.c The operators ∂/∂ z and ∂/∂ ¯z......Page 116
4.2.a Path integration......Page 118
4.2.b Integrals along a circle......Page 120
4.2.d Various forms of Cauchy’s theorem......Page 121
4.3.a The Cauchy formula and applications......Page 124
4.3.b Maximum modulus principle......Page 129
4.3.c Other theorems......Page 130
4.3.d Classification of zero sets of holomorphic functions......Page 131
4.4.a Classification of singularities......Page 133
4.4.b Meromorphic functions......Page 135
4.5.a Introduction and definition......Page 136
4.5.b Examples of Laurent series......Page 138
4.5.c The Residue theorem......Page 139
4.5.d Practical computations of residues......Page 141
4.6.a Jordan’s lemmas......Page 142
4.6.b Integrals on R of a rational function......Page 143
4.6.c Fourier integrals......Page 145
4.6.d Integral on the unit circle of a rational function......Page 146
4.6.e Computation of infinite sums......Page 147
5.1.a The complex logarithms......Page 160
5.1.c Multivalued functions, Riemann surfaces......Page 162
5.2.a Definitions......Page 164
5.2.b Properties......Page 165
5.2.c A trick to find f knowing u......Page 167
5.3 Analytic continuation......Page 169
5.4 Singularities at infinity......Page 171
5.5 The saddle point method......Page 173
5.5.a The general saddle point method......Page 174
5.5.b The real saddle point method......Page 177
6.1.a Preliminaries......Page 180
6.1.b The Riemann mapping theorem......Page 182
6.1.c Examples of conformal maps......Page 183
6.1.d The Schwarz-Christoffel transformation......Page 186
6.2 Applications to potential theory......Page 188
6.2.a Application to electrostatics......Page 190
6.2.b Application to hydrodynamics......Page 192
6.2.c Potential theory, lightning rods, and percolation......Page 194
6.3 Dirichlet problem and Poisson kernel......Page 195
7.1.a The problem of distribution of charge......Page 204
and forces during an elastic shock......Page 206
7.2 Definitions and examples of distributions......Page 207
7.2.a Regular distributions......Page 209
7.2.b Singular distributions......Page 210
7.2.d Other examples......Page 212
7.3.a Operations on distributions......Page 213
7.3.b Derivative of a distribution......Page 216
7.4.a The Heaviside distribution......Page 218
7.4.b Multidimensional Dirac distributions......Page 219
7.4.c The distribution δ′......Page 221
7.4.d Composition of δ with a function......Page 223
7.4.e Charge and current densities......Page 224
7.5.a Derivation of a function discontinuous at a point......Page 226
7.5.b Derivative of a function with discontinuity along a surface......Page 229
7.5.c Laplacian of a function discontinuous along a surface......Page 231
7.5.d Application: laplacian of 1/ r in 3-space......Page 232
7.6.b The tensor product of distributions......Page 234
7.6.c Convolution of two functions......Page 236
7.6.d “Fuzzy” measurement......Page 238
7.6.e Convolution of distributions......Page 239
7.6.f Applications......Page 240
7.6.g The Poisson equation......Page 241
7.7 Physical interpretation of convolution operators......Page 242
7.8 Discrete convolution......Page 245
8.1.a Definition......Page 248
8.1.b Application to the computation of certain integrals......Page 249
8.1.c Feynman’s notation......Page 250
8.1.d Kramers-Kronig relations......Page 252
8.1.e A few equations in the sense of distributions......Page 254
8.2.a Weak convergence in......Page 255
8.2.b Sequences of functions converging to δ......Page 256
8.2.d Regularization of a distribution......Page 259
8.2.e Continuity of convolution......Page 260
8.3 Convolution algebras......Page 261
8.4.a First order equations......Page 263
8.4.b The case of the harmonic oscillator......Page 264
8.4.c Other equations of physical origin......Page 265
9.1 Insufficiency of vector spaces......Page 274
9.2 Pre-Hilbert spaces......Page 276
9.2.b Projection on a finite-dimensional subspace......Page 279
9.3 Hilbert spaces......Page 281
9.3.a Hilbert basis......Page 282
9.3.b The ℓ2 space......Page 286
9.3.c The space L2 [0, a]......Page 287
9.3.d The L2(R) space......Page 288
9.4.a Fourier coefficients of a function......Page 289
9.4.b Mean-square convergence......Page 290
9.4.c Fourier series of a function f ∈ L1 [0, a]......Page 291
9.4.d Pointwise convergence of the Fourier series......Page 292
9.4.e Uniform convergence of the Fourier series......Page 294
9.4.f The Gibbs phenomenon......Page 295
10.1 Fourier transform of a function in L1......Page 302
10.1.a Definition......Page 303
10.1.c The L1 space......Page 304
10.1.d Elementary properties......Page 305
10.1.e Inversion......Page 307
10.1.f Extension of the inversion formula......Page 309
10.2.a Transpose and translates......Page 310
10.2.c Derivation......Page 311
10.3 Fourier transform of a function in L2......Page 313
10.3.a The space S......Page 314
10.3.b The Fourier transform in L2......Page 315
10.4.a Convolution formula......Page 317
10.4.b Cases of the convolution formula......Page 318
11.1 Definition and properties......Page 324
11.1.a Tempered distributions......Page 325
11.1.b Fourier transform of tempered distributions......Page 326
11.1.c Examples......Page 328
11.1.d Higher-dimensional Fourier transforms......Page 330
11.1.e Inversion formula......Page 331
11.2.a Definition and properties......Page 332
11.2.b Fourier transform of a periodic function......Page 333
11.2.c Poisson summation formula......Page 334
11.2.d Application to the computation of series......Page 335
11.3 The Gibbs phenomenon......Page 336
11.4.a Link between diaphragm and diffraction figure......Page 339
11.4.b Diaphragm made of infinitely many infinitely narrow slits......Page 340
11.4.c Finite number of infinitely narrow slits......Page 341
11.4.d Finitely many slits with finite width......Page 343
11.4.e Circular lens......Page 345
11.5 Limitations of Fourier analysis and wavelets......Page 346
12.1 Definition and integrability......Page 356
12.1.a Definition......Page 357
12.1.b Integrability......Page 358
12.2 Inversion......Page 361
12.3.a Translation......Page 363
12.3.c Differentiation and integration......Page 364
12.3.d Examples......Page 366
12.4.b Properties......Page 367
12.4.d The z-transform......Page 369
12.4.e Relation between Laplace and Fourier transforms......Page 370
12.5.a Importance of the Cauchy problem......Page 371
12.5.b A simple example......Page 372
12.5.c Dynamics of the electromagnetic field without sources......Page 373
13.1 Justification of sinusoidal regime analysis......Page 380
13.2 Fourier transform of vector fields: longitudinal and transverse fields......Page 383
13.3 Heisenberg uncertainty relations......Page 384
13.4 Analytic signals......Page 390
13.5.b Properties......Page 393
13.5.c Intercorrelation......Page 394
13.6.b Autocorrelation......Page 395
13.7 Application to optics: the Wiener-Khintchine theorem......Page 396
14.1.a Scalar product and representation theorem......Page 402
14.1.b Adjoint......Page 403
14.2.a Kets |ψ> ∈ H......Page 404
14.2.b Bras <ψ| ∈ H′......Page 405
14.2.c Generalized bras......Page 407
14.2.d Generalized kets......Page 408
14.2.e Id =......Page 409
14.2.f Generalized basis......Page 410
14.3.a Operators......Page 412
14.3.b Adjoint......Page 414
14.3.c Bounded operators, closed operators, closable operators......Page 415
14.3.d Discrete and continuous spectra......Page 416
14.4 Hermitian operators; self-adjoint operators......Page 418
14.4.a Definitions......Page 419
14.4.b Eigenvectors......Page 421
14.4.c Generalized eigenvectors......Page 422
14.4.d “Matrix” representation......Page 423
14.4.e Summary of properties of the operators P and X......Page 426
15.1 Generalities about Green functions......Page 432
15.2 A pedagogical example: the harmonic oscillator......Page 434
15.2.b Using the Fourier transform......Page 435
15.3.a Computation of the advanced and retarded Green functions......Page 439
15.3.b Retarded potentials......Page 443
15.3.d Radiation......Page 446
15.4 The heat equation......Page 447
15.4.a One-dimensional case......Page 448
15.4.b Three-dimensional case......Page 451
15.5 Quantum mechanics......Page 452
15.6 Klein-Gordon equation......Page 454
16.1.a Vectors......Page 458
16.1.b Einstein convention......Page 460
16.1.c Linear forms......Page 461
16.1.d Linear maps......Page 463
16.2.a Existence of the tensor product of two vector spaces......Page 464
16.2.b Tensor product of linear forms: tensors of type......Page 466
16.2.c Tensor product of vectors: tensors of type......Page 468
16.2.d Tensor product of a vector and a linear form: linear -tensorsmapsor......Page 469
16.2.e Tensors of type......Page 471
16.3.a Metric and pseudo-metric......Page 472
16.3.b Natural duality by means of the metric......Page 474
16.3.c Gymnastics: raising and lowering indices......Page 475
16.4 Operations on tensors......Page 478
16.5.a Curvilinear coordinates......Page 480
16.5.b Basis vectors......Page 481
16.5.c Transformation of physical quantities......Page 483
16.5.d Transformation of linear forms......Page 484
16.5.e Transformation of an arbitrary tensor field......Page 485
16.5.f Conclusion......Page 486
17.1.a 1-forms......Page 488
17.1.b Exterior 2-forms......Page 489
17.1.c Exterior k-forms......Page 490
17.1.d Exterior product......Page 492
17.2.a Definition......Page 494
17.2.b Exterior derivative......Page 495
17.3 Integration of differential forms......Page 496
17.4 Poincar´e’s theorem......Page 499
17.5.a Differential forms in dimension 3......Page 501
17.5.b Existence of the scalar electrostatic potential......Page 502
17.5.c Existence of the vector potential......Page 504
17.6 Electromagnetism in the language of differential forms......Page 505
18.1 Groups......Page 514
18.2 Linear representations of groups......Page 516
18.3 Vectors and the group SO(3)......Page 517
18.4 The group SU(2) and spinors......Page 522
18.5 Spin and Riemann sphere......Page 528
19 Introduction to probability theory......Page 534
19.1 Introduction......Page 535
19.2 Basic definitions......Page 537
19.3 Poincar´e formula......Page 541
19.4 Conditional probability......Page 542
19.5 Independent events......Page 544
20.1 Random variables and probability distributions......Page 546
20.2 Distribution function and probability density......Page 549
20.2.b (Absolutely) continuous random variables......Page 551
20.3.a Case of a discrete r.v.......Page 552
20.3.b Case of a continuous r.v.......Page 553
20.4.a Particles in a confined gas......Page 555
20.4.b Radioactive decay......Page 556
20.5 Moments of a random variable......Page 557
20.6.a Pair of random variables......Page 559
20.6.b Independent random variables......Page 562
20.6.c Random vectors......Page 563
20.7.a Case of a single random variable......Page 564
20.8.a Expectation of a function of random variables......Page 565
20.8.c Characteristic function......Page 566
20.9.a Sum of random variables......Page 568
20.9.b Product of random variables......Page 571
20.10.a Statement......Page 572
20.10.b Application: Buffon’s needle......Page 574
Independance, correlation, causality......Page 575
21.1 Various types of convergence......Page 578
21.2 The law of large numbers......Page 580
21.3 Central limit theorem......Page 581
A.1 Topology, topological spaces......Page 598
A.2.a Norms, seminorms......Page 602
A.2.b Balls and topology associated to the distance......Page 603
A.2.c Comparison of sequences......Page 605
A.2.e Comparison of norms......Page 606
A.2.f Norm of a linear map......Page 608
B.1.a Functions of one real variable......Page 610
B.1.b Differential of a function f : R......Page 611
B.2 Differential of map with values in R......Page 612
B.3 Lagrange multipliers......Page 613
C.1 Duality......Page 618
C.2.b Matrix of a linear map......Page 619
C.2.d Change of basis formula......Page 620
C.2.e Case of an orthonormal basis......Page 621
Appendic D: A few proofs......Page 622
Tables......Page 632
Fourier Transforms......Page 634
Three-dimensional Fourier transforms......Page 636
Table of the usual conventions for the Fourier transform......Page 637
Laplace transforms......Page 638
Tables of probability laws......Page 641
References......Page 646
Portraits......Page 652
Sidebars......Page 654
Index......Page 656
Further reading......Page 642