1 Reminders: convergence of sequences and series 1
1.1 The problem of limits in physics 1
1.1.a Two paradoxes involving kinetic energy 1
1.1.b Romeo, Juliet, and viscous fluids 5
1.1.c Potential wall in quantum mechanics 7
1.1.d Semi-infinite filter behaving as waveguide 9
1.2 Sequences 12
1.2.a Sequences in a normed vector space 12
1.2.b Cauchy sequences 13
1.2.c The fixed point theorem 15
1.2.d Double sequences 16
1.2.e Sequential definition of the limit of a function 17
1.2.f Sequences of functions 18
1.3 Series 23
1.3.a Series in a normed vector space 23
1.3.b Doubly infinite series 24
1.3.c Convergence of a double series 25
1.3.d Conditionally convergent series, absolutely convergent series. 26
1.3.e Series of functions 29
1.4 Power series, analytic functions 30
1.4.a Taylor formulas 31
1.4.b Some numerical illustrations 32
1.4.c Radius of convergence of a power series 34
1.4.d Analytic functions 35
1.5 A quick look at asymptotic and divergent series 37
1.5.a Asymptotic series 37
1.5.b Divergent series and asymptotic expansions 38
Exercises 43
Problem 46
Solutions 47
2 Measure theory and the Lebesgue integral 51
2.1 The integral according to Mr. Riemann 51
2.1.a Riemann sums 51
2.1.b Limitations of Riemann's definition 54
2.2 The integral according to Mr. Lebesgue 54
2.2.a Principle of the method 55
2.2.b Borel subsets 57
2.2.c Lebesgue measure 59
2.2.d The Lebesgue σ-algebra 60
2.2.e Negligible sets 61
2.2.f Lebesgue measure on ？ 62
2.2.g Definition of the Lebesgue integral 63
2.2.h Functions zero almost everywhere, space L1 66
2.2.i And today? 67
Exercises 68
Solutions 71
3 Integral calculus 73
3.1 Integrability in practice 73
3.1.a Standard functions 73
3.1.b Comparison theorems 74
3.2 Exchanging integrals and limits or series 75
3.3 Integrals with parameters 77
3.3.a Continuity of functions defined by integrals 77
3.3.b Differentiating under the integral sign 78
3.3.c Case of parameters appearing in the integration range 78
3.4 Double and multiple integrals 79
3.5 Change of variables 81
Exercises 83
Solutions 85
4 Complex Analysis Ⅰ 87
4.1 Holomorphic functions 87
4.1.a Definitions 88
4.1.b Examples 90
4.1.c The operators ？/？z and ？/？？ 91
4.2 Cauchy's theorem 93
4.2.a Path integration 93
4.2.b Integrals along a circle 95
4.2.c Winding number 96
4.2.d Various forms of Cauchy's theorem 96
4.2.e Application 99
4.3 Properties of holomorphic functions 99
4.3.a The Cauchy formula and applications 99
4.3.b Maximum modulus principle 104
4.3.c Other theorems 105
4.3.d Classification of zero sets of holomorphic functions 106
4.4 Singularities of a function 108
4.4.a Classification of singularities 108
4.4.b Meromorphic functions 110
4.5 Laurent series 111
4.5.a Introduction and definition 111
4.5.b Examples of Laurent series 113
4.5.c The Residue theorem 114
4.5.d Practical computations of residues 116
4.6 Applications to the computation of horrifying integrals or ghastly sums 117
4.6.a Jordan's lemmas 117
4.6.b Integrals on ？ of a rational function 118
4.6.c Fourier integrals 120
4.6.d Integral on the unit circle of a rational function 121
4.6.e Computation of infinite sums 122
Exercises 125
Problem 128
Solutions 129
5 Complex Analysis Ⅱ 135
5.1 Complex logarithm; multivalued functions 135
5.1.a The complex logarithms 135
5.1.b The square root function 137
5.1.c Multivalued functions, Riemann surfaces 137
5.2 Harmonic functions 139
5.2.a Definitions 139
5.2.b Properties 140
5.2.c A trick to find f knowing u 142
5.3 Analytic continuation 144
5.4 Singularities at infinity 146
5.5 The saddle point method 148
5.5.a The general saddle point method 149
5.5.b The real saddle point method 152
Exercises 153
Solutions 154
6 Conformal maps 155
6.1 Conformal maps 155
6.1.a Preliminaries 155
6.1.b The Riemann mapping theorem 157
6.1.c Examples of conformal maps 158
6.1.d The Schwarz-Christoffel transformation 161
6.2 Applications to potential theory 163
6.2.a Application to electrostatics 165
6.2.b Application to hydrodynamics 167
6.2.c Potential theory, lightning rods, and percolation 169
6.3 Dirichlet problem and Poisson kernel 170
Exercises 174
Solutions 176
7 Distributions Ⅰ 179
7.1 Physical approach 179
7.1.a The problem of distribution of charge 179
7.1.b The problem of momentum and forces during an elastic shock 181
7.2 Definitions and examples of distributions 182
7.2.a Regular distributions 184
7.2.b Singular distributions 185
7.2.c Support of a distribution 187
7.2.d Other examples 187
7.3 Elementary properties. Operations 188
7.3.a Operations on distributions 188
7.3.b Derivative of a distribution 191
7.4 Dirac and its derivatives 193
7.4.a The Heaviside distribution 193
7.4.b Multidimensional Dirac distributions 194
7.4.c The distribution δ′ 196
7.4.d Composition of δ with a function 198
7.4.e Charge and current densities 199
7.5 Derivation of a discontinuous function 201
7.5.a Derivation of a function discontinuous at a point 201
7.5.b Derivative of a function with discontinuity along a surface ？ 204
7.5.c Laplacian of a function discontinuous along a surface ？ 206
7.5.d Application: laplacian of 1/r in 3-space 207
7.6 Convolution 209
7.6.a The tensor product of two functions 209
7.6.b The tensor product of distributions 209
7.6.c Convolution of two functions 211
7.6.d "Fuzzy" measurement 213
7.6.e Convolution of distributions 214
7.6.f Applications 215
7.6.g The Poisson equation 216
7.7 Physical interpretation of convolution operators 217
7.8 Discrete convolution 220
8 Distributions Ⅱ 223
8.1 Cauchy principal value 223
8.1.a Definition 223
8.1.b Application to the computation of certain integrals 224
8.1.c Feynman's notation 225
8.1.d Kramers-Kronig relations 227
8.1.e A few equations in the sense of distributions 229
8.2 Topology in ？ 230
8.2.a Weak convergence in ？ 230
8.2.b Sequences of functions converging to δ 231
8.2.c Convergence in ？ and convergence in the sense of functions 234
8.2.d Regularization of a distribution 234
8.2.e Continuity of convolution 235
8.3 Convolution algebras 236
8.4 Solving a differential equation with initial conditions 238
8.4.a First order equations 238
8.4.b The case of the harmonic oscillator 239
8.4.c Other equations of physical origin 240
Exercises 241
Problem 244
Solutions 245
9 Hilbert spaces; Fourier series 249
9.1 Insufficiency of vector spaces 249
9.2 Pre-Hilbert spaces 251
9.2.a The finite-dimensional case 254
9.2.b Projection on a finite-dimensional subspace 254
9.2.c Bessel inequality 256
9.3 Hilbert spaces 256
9.3.a Hilbert basis 257
9.3.b The e2 space 261
9.3.c The space L2 [0,a] 262
9.3.d The L2(？) space 263
9.4 Fourier series expansion 264
9.4.a Fourier coefficients of a function 264
9.4.b Mean-square convergence 265
9.4.c Fourier series of a function f ∈ L1 [0,a] 266
9.4.d Pointwise convergence of the Fourier series 267
9.4.e Uniform convergence of the Fourier series 269
9.4.f The Gibbs phenomenon 270
Exercises 270
Problem 271
Solutions 272
10 Fourier transform of functions 277
10.1 Fourier transform of a function in L1 277
10.1.a Definition 278
10.1.b Examples 279
10.1.c The L1 space 279
10.1.d Elementary properties 280
10.1.e Inversion 282
10.1.f Extension of the inversion formula 284
10.2 Properties of the Fourier transform 285
10.2.a Transpose and translates 285
10.2.b Dilation 286
10.2.c Derivation 286
10.2.d Rapidly decaying functions 288
10.3 Fourier transform of a function in L2 288
10.3.a The space ？ 289
10.3.b The Fourier transform in L2 290
10.4 Fourier transform and convolution 292
10.4.a Convolution formula 292
10.4.b Cases of the convolution formula 293
Exercises 295
Solutions 296
11 Fourier transform of distributions 299
11.1 Definition and properties 299
11.1.a Tempered distributions 300
11.1.b Fourier transform of tempered distributions 301
11.1.c Examples 303
11.1.d Higher-dimensional Fourier transforms 305
11.1.e Inversion formula 306
11.2 The Dirac comb 307
11.2.a Definition and properties 307
11.2.b Fourier transform of a periodic function 308
11.2.c Poisson summation formula 309
11.2.d Application to the computation of series 310
11.3 The Gibbs phenomenon 311
11.4 Application to physical optics 314
11.4.a Link between diaphragm and diffraction figure 314
11.4.b Diaphragm made of infinitely many infinitely narrow slits 315
11.4.c Finite number of infinitely narrow slits 316
11.4.d Finitely many slits with finite width 318
11.4.e Circular lens 320
11.5 Limitations of Fourier analysis and wavelets 321
Exercises 324
Problem 325
Solutions 326
12 The Laplace transform 331
12.1 Definition and integrability 331
12.1.a Definition 332
12.1.b Integrability 333
12.1.c Properties of the Laplace transform 336
12.2 Inversion 336
12.3 Elementary properties and examples of Laplace transforms 338
12.3.a Translation 338
12.3.b Convolution 339
12.3.c Differentiation and integration 339
12.3.d Examples 341
12.4 Laplace transform of distributions 342
12.4.a Definition 342
12.4.b Properties 342
12.4.c Examples 344
12.4.d The z-transform 344
12.4.e Relation between Laplace and Fourier transforms 345
12.5 Physical applications, the Cauchy problem 346
12.5.a Importance of the Cauchy problem 346
12.5.b A simple example 347
12.5.c Dynamics of the electromagnetic field without sources 348
Exercises 351
Solutions 352
13 Physical applications of the Fourier transform 355
13.1 Justification of sinusoidal regime analysis 355
13.2 Fourier transform of vector fields: longitudinal and transverse fields 358
13.3 Heisenberg uncertainty relations 359
13.4 Analytic signals 365
13.5 Autocorrelation of a finite energy function 368
13.5.a Definition 368
13.5.b Properties 368
13.5.c Intercorrelation 369
13.6 Finite power functions 370
13.6.a Definitions 370
13.6.b Autocorrelation 370
13.7 Application to optics: the Wiener-Khintchine theorem 371
Exercises 375
Solutions 376
14 Bras, kets, and all that sort of thing 377
14.1 Reminders about finite dimension 377
14.1.a Scalar product and representation theorem 377
14.1.b Adjoint 378
14.1.c Symmetric and hermitian endomorphisms 379
14.2 Kets and bras 379
14.2.a Kets ？> ∈ H 379
14.2.b Bras <？ ∈ H′ 380
14.2.c Generalized bras 382
14.2.d Generalized kets 383
14.2.e Id = ∑n ？n> <？n？ 384
14.2.f Generalized basis 385
14.3 Linear operators 387
14.3.a Operators 387
14.3.b Adjoint 389
14.3.c Bounded operators, closed operators, closable operators 390
14.3.d Discrete and continuous spectra 391
14.4 Hermitian operators; self-adjoint operators 393
14.4.a Definitions 394
14.4.b Eigenvectors 396
14.4.c Generalized eigenvectors 397
14.4.d "Matrix" representation 398
14.4.e Summary of properties of the operators P and X 401
Exercises 403
Solutions 404
15 Green functions 407
15.1 Generalities about Green functions 407
15.2 A pedagogical example: the harmonic oscillator 409
15.2.a Using the Laplace transform 410
15.2.b Using the Fourier transform 410
15.3 Electromagnetism and the d'Alembertian operator 414
15.3.a Computation of the advanced and retarded Green functions 414
15.3.b Retarded potentials 418
15.3.c Covariant expression of advanced and retarded Green functions 421
15.3.d Radiation 421
15.4 The heat equation 422
15.4.a One-dimensional case 423
15.4.b Three-dimensional case 426
15.5 Quantum mechanics 427
15.6 Klein-Gordon equation 429
Exercises 432
16 Tensors 433
16.1 Tensors in affine space 433
16.1.a Vectors 433
16.1.b Einstein convention 435
16.1.c Linear forms 436
16.1.d Linear maps 438
16.1.e Lorentz transformations 439
16.2 Tensor product of vector spaces: tensors 439
16.2.a Existence of the tensor product of two vector spaces 439
16.2.b Tensor product of linear forms: tensors of type (0 2) 441
16.2.c Tensor product of vectors: tensors of type (2 0) 443
16.2.d Tensor product of a vector and a linear form: linear maps or (1 1)-tensors 444
16.2.e Tensors of type (p q) 446
16.3 The metric, or, how to raise and lower indices 447
16.3.a Metric and pseudo-metric 447
16.3.b Natural duality by means of the metric 449
16.3.c Gymnastics: raising and lowering indices 450
16.4 Operations on tensors 453
16.5 Change of coordinates 455
16.5.a Curvilinear coordinates 455
16.5.b Basis vectors 456
16.5.c Transformation of physical quantities 458
16.5.d Transformation of linear forms 459
16.5.e Transformation of an arbitrary tensor field 460
16.5.f Conclusion 461
Solutions 462
17 Differential forms 463
17.1 Exterior algebra 463
17.1.a 1-forms 463
17.1.b Exterior 2-forms 464
17.1.c Exterior k-forms 465
17.1.d Exterior product 467
17.2 Differential forms on a vector space 469
17.2.a Definition 469
17.2.b Exterior derivative 470
17.3 Integration of differential forms 471
17.4 Poincaré's theorem 474
17.5 Relations with vector calculus: gradient, divergence, curl 476
17.5.a Differential forms in dimension 3 476
17.5.b Existence of the scalar electrostatic potential 477
17.5.c Existence of the vector potential 479
17.5.d Magnetic monopoles 480
17.6 Electromagnetism in the language of differential forms 480 484
Problem 485
Solution 489
18 Groups and group representations 489
18.1 Groups 489
18.2 Linear representations of groups 491
18.3 Vectors and the group SO(3) 492
18.4 The group SU(2) and spinors 497
18.5 Spin and Riemann sphere 503
Exercises 505
19 Introduction to probability theory 509
19.1 Introduction 510
19.2 Basic definitions 512
19.3 Poincaré formula 516
19.4 Conditional probability 517
19.5 Independent events 519
20 Random variables 521
20.1 Random variables and probability distributions 521
20.2 Distribution function and probability density 524
20.2.a Discrete random variables 526
20.2.b (Absolutely) continuous random variables 526
20.3 Expectation and variance 527
20.3.a Case of a discrete r.v. 527
20.3.b Case of a continuous r.v. 528
20.4 An example: the Poisson distribution 530
20.4.a Particles in a confined gas 530
20.4.b Radioactive decay 531
20.5 Moments of a random variable 532
20.6 Random vectors 534
20.6.a Pair of random variables 534
20.6.b Independent random variables 537
20.6.c Random vectors 538
20.7 Image measures 539
20.7.a Case of a single random variable 539
20.7.b Case of a random vector 540
20.8 Expectation and characteristic function 540
20.8.a Expectation of a function of random variables 540
20.8.b Moments, variance 541
20.8.c Characteristic function 541
20.8.d Generating function 543
20.9 Sum and product of random variables 543
20.9.a Sum of random variables 543
20.9.b Product of random variables 546
20.9.c Example: Poisson distribution 547
20.10 Bienaymé-Tchebychev inequality 547
20.10.a Statement 547
20.10.b Application: Buffon's needle 549
20.11 Independance, correlation, causality 550
21 Convergence of random variables: central limit theorem 553
21.1 Various types of convergence 553
21.2 The law of large numbers 555
21.3 Central limit theorem 556
Exercises 560
Problems 563
Solutions 564
Appendices 573
A Reminders concerning topology and normed vector spaces 573
A.1 Topology, topological spaces 573
A.2 Normed vector spaces 577
A.2.a Norms, seminorms 577
A.2.b Balls and topology associated to the distance 578
A.2.c Comparison of sequences 580
A.2.d Bolzano-Weierstrass theorems 581
A.2.e Comparison of norms 581
A.2.f Norm of a linear map 583
Exercise 583
Solution 584
B Elementary reminders of differential calculus 585
B.1 Differential of a real-valued function 585
B.1.a Functions of one real variable 585
B.1.b Differential of a function f : ？n → ？ 586
B.1.c Tensor notation 587
B.2 Differential of map with values in ？p 587
B.3 Lagrange multipliers 588
Solution 591
C Matrices 593
C.1 Duality 593
C.2 Application to matrix representation 594
C.2.a Matrix representing a family of vectors 594
C.2.b Matrix of a linear map 594
C.2.c Change of basis 595
C.2.d Change of basis formula 595
C.2.e Case of an orthonormal basis 596
D A few proofs 597
Tables 609
Fourier transforms 609
Laplace transforms 613
Probability laws 616
Further reading 617
References 621
Portraits 627
Sidebars 629
Index 631