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1 Reminders: convergence of sequences and series1

1.1 The problem of limits in physics1

1.1.a Two paradoxes involving kinetic energy1

1.1.b Romeo, Juliet, and viscous fluids5

1.1.c Potential wall in quantum mechanics7

1.1.d Semi-infinite filter behaving as waveguide9

1.2 Sequences12

1.2.a Sequences in a normed vector space12

1.2.b Cauchy sequences13

1.2.c The fixed point theorem15

1.2.d Double sequences16

1.2.e Sequential definition of the limit of a function17

1.2.f Sequences of functions18

1.3 Series23

1.3.a Series in a normed vector space23

1.3.b Doubly infinite series24

1.3.c Convergence of a double series25

1.3.d Conditionally convergent series, absolutely convergent series.26

1.3.e Series of functions29

1.4 Power series, analytic functions30

1.4.a Taylor formulas31

1.4.b Some numerical illustrations32

1.4.c Radius of convergence of a power series34

1.4.d Analytic functions35

1.5 A quick look at asymptotic and divergent series37

1.5.a Asymptotic series37

1.5.b Divergent series and asymptotic expansions38

Exercises43

Problem46

Solutions47

2 Measure theory and the Lebesgue integral51

2.1 The integral according to Mr. Riemann51

2.1.a Riemann sums51

2.1.b Limitations of Riemann's definition54

2.2 The integral according to Mr. Lebesgue54

2.2.a Principle of the method55

2.2.b Borel subsets57

2.2.c Lebesgue measure59

2.2.d The Lebesgue σ-algebra60

2.2.e Negligible sets61

2.2.f Lebesgue measure on ？62

2.2.g Definition of the Lebesgue integral63

2.2.h Functions zero almost everywhere, space L166

2.2.i And today?67

Exercises68

Solutions71

3 Integral calculus73

3.1 Integrability in practice73

3.1.a Standard functions73

3.1.b Comparison theorems74

3.2 Exchanging integrals and limits or series75

3.3 Integrals with parameters77

3.3.a Continuity of functions defined by integrals77

3.3.b Differentiating under the integral sign78

3.3.c Case of parameters appearing in the integration range78

3.4 Double and multiple integrals79

3.5 Change of variables81

Exercises83

Solutions85

4 Complex Analysis Ⅰ87

4.1 Holomorphic functions87

4.1.a Definitions88

4.1.b Examples90

4.1.c The operators ？/？z and ？/？？91

4.2 Cauchy's theorem93

4.2.a Path integration93

4.2.b Integrals along a circle95

4.2.c Winding number96

4.2.d Various forms of Cauchy's theorem96

4.2.e Application99

4.3 Properties of holomorphic functions99

4.3.a The Cauchy formula and applications99

4.3.b Maximum modulus principle104

4.3.c Other theorems105

4.3.d Classification of zero sets of holomorphic functions106

4.4 Singularities of a function108

4.4.a Classification of singularities108

4.4.b Meromorphic functions110

4.5 Laurent series111

4.5.a Introduction and definition111

4.5.b Examples of Laurent series113

4.5.c The Residue theorem114

4.5.d Practical computations of residues116

4.6 Applications to the computation of horrifying integrals or ghastly sums117

4.6.a Jordan's lemmas117

4.6.b Integrals on ？ of a rational function118

4.6.c Fourier integrals120

4.6.d Integral on the unit circle of a rational function121

4.6.e Computation of infinite sums122

Exercises125

Problem128

Solutions129

5 Complex Analysis Ⅱ135

5.1 Complex logarithm; multivalued functions135

5.1.a The complex logarithms135

5.1.b The square root function137

5.1.c Multivalued functions, Riemann surfaces137

5.2 Harmonic functions139

5.2.a Definitions139

5.2.b Properties140

5.2.c A trick to find f knowing u142

5.3 Analytic continuation144

5.4 Singularities at infinity146

5.5 The saddle point method148

5.5.a The general saddle point method149

5.5.b The real saddle point method152

Exercises153

Solutions154

6 Conformal maps155

6.1 Conformal maps155

6.1.a Preliminaries155

6.1.b The Riemann mapping theorem157

6.1.c Examples of conformal maps158

6.1.d The Schwarz-Christoffel transformation161

6.2 Applications to potential theory163

6.2.a Application to electrostatics165

6.2.b Application to hydrodynamics167

6.2.c Potential theory, lightning rods, and percolation169

6.3 Dirichlet problem and Poisson kernel170

Exercises174

Solutions176

7 Distributions Ⅰ179

7.1 Physical approach179

7.1.a The problem of distribution of charge179

7.1.b The problem of momentum and forces during an elastic shock181

7.2 Definitions and examples of distributions182

7.2.a Regular distributions184

7.2.b Singular distributions185

7.2.c Support of a distribution187

7.2.d Other examples187

7.3 Elementary properties. Operations188

7.3.a Operations on distributions188

7.3.b Derivative of a distribution191

7.4 Dirac and its derivatives193

7.4.a The Heaviside distribution193

7.4.b Multidimensional Dirac distributions194

7.4.c The distribution δ′196

7.4.d Composition of δ with a function198

7.4.e Charge and current densities199

7.5 Derivation of a discontinuous function201

7.5.a Derivation of a function discontinuous at a point201

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-space207

7.6 Convolution209

7.6.a The tensor product of two functions209

7.6.b The tensor product of distributions209

7.6.c Convolution of two functions211

7.6.d "Fuzzy" measurement213

7.6.e Convolution of distributions214

7.6.f Applications215

7.6.g The Poisson equation216

7.7 Physical interpretation of convolution operators217

7.8 Discrete convolution220

8 Distributions Ⅱ223

8.1 Cauchy principal value223

8.1.a Definition223

8.1.b Application to the computation of certain integrals224

8.1.c Feynman's notation225

8.1.d Kramers-Kronig relations227

8.1.e A few equations in the sense of distributions229

8.2 Topology in ？230

8.2.a Weak convergence in ？230

8.2.b Sequences of functions converging to δ231

8.2.c Convergence in ？ and convergence in the sense of functions234

8.2.d Regularization of a distribution234

8.2.e Continuity of convolution235

8.3 Convolution algebras236

8.4 Solving a differential equation with initial conditions238

8.4.a First order equations238

8.4.b The case of the harmonic oscillator239

8.4.c Other equations of physical origin240

Exercises241

Problem244

Solutions245

9 Hilbert spaces; Fourier series249

9.1 Insufficiency of vector spaces249

9.2 Pre-Hilbert spaces251

9.2.a The finite-dimensional case254

9.2.b Projection on a finite-dimensional subspace254

9.2.c Bessel inequality256

9.3 Hilbert spaces256

9.3.a Hilbert basis257

9.3.b The e2 space261

9.3.c The space L2 [0,a]262

9.3.d The L2(？) space263

9.4 Fourier series expansion264

9.4.a Fourier coefficients of a function264

9.4.b Mean-square convergence265

9.4.c Fourier series of a function f ∈ L1 [0,a]266

9.4.d Pointwise convergence of the Fourier series267

9.4.e Uniform convergence of the Fourier series269

9.4.f The Gibbs phenomenon270

Exercises270

Problem271

Solutions272

10 Fourier transform of functions277

10.1 Fourier transform of a function in L1277

10.1.a Definition278

10.1.b Examples279

10.1.c The L1 space279

10.1.d Elementary properties280

10.1.e Inversion282

10.1.f Extension of the inversion formula284

10.2 Properties of the Fourier transform285

10.2.a Transpose and translates285

10.2.b Dilation286

10.2.c Derivation286

10.2.d Rapidly decaying functions288

10.3 Fourier transform of a function in L2288

10.3.a The space ？289

10.3.b The Fourier transform in L2290

10.4 Fourier transform and convolution292

10.4.a Convolution formula292

10.4.b Cases of the convolution formula293

Exercises295

Solutions296

11 Fourier transform of distributions299

11.1 Definition and properties299

11.1.a Tempered distributions300

11.1.b Fourier transform of tempered distributions301

11.1.c Examples303

11.1.d Higher-dimensional Fourier transforms305

11.1.e Inversion formula306

11.2 The Dirac comb307

11.2.a Definition and properties307

11.2.b Fourier transform of a periodic function308

11.2.c Poisson summation formula309

11.2.d Application to the computation of series310

11.3 The Gibbs phenomenon311

11.4 Application to physical optics314

11.4.a Link between diaphragm and diffraction figure314

11.4.b Diaphragm made of infinitely many infinitely narrow slits315

11.4.c Finite number of infinitely narrow slits316

11.4.d Finitely many slits with finite width318

11.4.e Circular lens320

11.5 Limitations of Fourier analysis and wavelets321

Exercises324

Problem325

Solutions326

12 The Laplace transform331

12.1 Definition and integrability331

12.1.a Definition332

12.1.b Integrability333

12.1.c Properties of the Laplace transform336

12.2 Inversion336

12.3 Elementary properties and examples of Laplace transforms338

12.3.a Translation338

12.3.b Convolution339

12.3.c Differentiation and integration339

12.3.d Examples341

12.4 Laplace transform of distributions342

12.4.a Definition342

12.4.b Properties342

12.4.c Examples344

12.4.d The z-transform344

12.4.e Relation between Laplace and Fourier transforms345

12.5 Physical applications, the Cauchy problem346

12.5.a Importance of the Cauchy problem346

12.5.b A simple example347

12.5.c Dynamics of the electromagnetic field without sources348

Exercises351

Solutions352

13 Physical applications of the Fourier transform355

13.1 Justification of sinusoidal regime analysis355

13.2 Fourier transform of vector fields: longitudinal and transverse fields358

13.3 Heisenberg uncertainty relations359

13.4 Analytic signals365

13.5 Autocorrelation of a finite energy function368

13.5.a Definition368

13.5.b Properties368

13.5.c Intercorrelation369

13.6 Finite power functions370

13.6.a Definitions370

13.6.b Autocorrelation370

13.7 Application to optics: the Wiener-Khintchine theorem371

Exercises375

Solutions376

14 Bras, kets, and all that sort of thing377

14.1 Reminders about finite dimension377

14.1.a Scalar product and representation theorem377

14.1.b Adjoint378

14.1.c Symmetric and hermitian endomorphisms379

14.2 Kets and bras379

14.2.a Kets ？> ∈ H379

14.2.b Bras <？ ∈ H′380

14.2.c Generalized bras382

14.2.d Generalized kets383

14.2.e Id = ∑n ？n> <？n？384

14.2.f Generalized basis385

14.3 Linear operators387

14.3.a Operators387

14.3.b Adjoint389

14.3.c Bounded operators, closed operators, closable operators390

14.3.d Discrete and continuous spectra391

14.4 Hermitian operators; self-adjoint operators393

14.4.a Definitions394

14.4.b Eigenvectors396

14.4.c Generalized eigenvectors397

14.4.d "Matrix" representation398

14.4.e Summary of properties of the operators P and X401

Exercises403

Solutions404

15 Green functions407

15.1 Generalities about Green functions407

15.2 A pedagogical example: the harmonic oscillator409

15.2.a Using the Laplace transform410

15.2.b Using the Fourier transform410

15.3 Electromagnetism and the d'Alembertian operator414

15.3.a Computation of the advanced and retarded Green functions414

15.3.b Retarded potentials418

15.3.c Covariant expression of advanced and retarded Green functions421

15.3.d Radiation421

15.4 The heat equation422

15.4.a One-dimensional case423

15.4.b Three-dimensional case426

15.5 Quantum mechanics427

15.6 Klein-Gordon equation429

Exercises432

16 Tensors433

16.1 Tensors in affine space433

16.1.a Vectors433

16.1.b Einstein convention435

16.1.c Linear forms436

16.1.d Linear maps438

16.1.e Lorentz transformations439

16.2 Tensor product of vector spaces: tensors439

16.2.a Existence of the tensor product of two vector spaces439

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)-tensors444

16.2.e Tensors of type (p q)446

16.3 The metric, or, how to raise and lower indices447

16.3.a Metric and pseudo-metric447

16.3.b Natural duality by means of the metric449

16.3.c Gymnastics: raising and lowering indices450

16.4 Operations on tensors453

16.5 Change of coordinates455

16.5.a Curvilinear coordinates455

16.5.b Basis vectors456

16.5.c Transformation of physical quantities458

16.5.d Transformation of linear forms459

16.5.e Transformation of an arbitrary tensor field460

16.5.f Conclusion461

Solutions462

17 Differential forms463

17.1 Exterior algebra463

17.1.a 1-forms463

17.1.b Exterior 2-forms464

17.1.c Exterior k-forms465

17.1.d Exterior product467

17.2 Differential forms on a vector space469

17.2.a Definition469

17.2.b Exterior derivative470

17.3 Integration of differential forms471

17.4 Poincaré's theorem474

17.5 Relations with vector calculus: gradient, divergence, curl476

17.5.a Differential forms in dimension 3476

17.5.b Existence of the scalar electrostatic potential477

17.5.c Existence of the vector potential479

17.5.d Magnetic monopoles480

17.6 Electromagnetism in the language of differential forms 480484

Problem485

Solution489

18 Groups and group representations489

18.1 Groups489

18.2 Linear representations of groups491

18.3 Vectors and the group SO(3)492

18.4 The group SU(2) and spinors497

18.5 Spin and Riemann sphere503

Exercises505

19 Introduction to probability theory509

19.1 Introduction510

19.2 Basic definitions512

19.3 Poincaré formula516

19.4 Conditional probability517

19.5 Independent events519

20 Random variables521

20.1 Random variables and probability distributions521

20.2 Distribution function and probability density524

20.2.a Discrete random variables526

20.2.b (Absolutely) continuous random variables526

20.3 Expectation and variance527

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 distribution530

20.4.a Particles in a confined gas530

20.4.b Radioactive decay531

20.5 Moments of a random variable532

20.6 Random vectors534

20.6.a Pair of random variables534

20.6.b Independent random variables537

20.6.c Random vectors538

20.7 Image measures539

20.7.a Case of a single random variable539

20.7.b Case of a random vector540

20.8 Expectation and characteristic function540

20.8.a Expectation of a function of random variables540

20.8.b Moments, variance541

20.8.c Characteristic function541

20.8.d Generating function543

20.9 Sum and product of random variables543

20.9.a Sum of random variables543

20.9.b Product of random variables546

20.9.c Example: Poisson distribution547

20.10 Bienaymé-Tchebychev inequality547

20.10.a Statement547

20.10.b Application: Buffon's needle549

20.11 Independance, correlation, causality550

21 Convergence of random variables: central limit theorem553

21.1 Various types of convergence553

21.2 The law of large numbers555

21.3 Central limit theorem556

Exercises560

Problems563

Solutions564

Appendices573

A Reminders concerning topology and normed vector spaces573

A.1 Topology, topological spaces573

A.2 Normed vector spaces577

A.2.a Norms, seminorms577

A.2.b Balls and topology associated to the distance578

A.2.c Comparison of sequences580

A.2.d Bolzano-Weierstrass theorems581

A.2.e Comparison of norms581

A.2.f Norm of a linear map583

Exercise583

Solution584

B Elementary reminders of differential calculus585

B.1 Differential of a real-valued function585

B.1.a Functions of one real variable585

B.1.b Differential of a function f : ？n → ？586

B.1.c Tensor notation587

B.2 Differential of map with values in ？p587

B.3 Lagrange multipliers588

Solution591

C Matrices593

C.1 Duality593

C.2 Application to matrix representation594

C.2.a Matrix representing a family of vectors594

C.2.b Matrix of a linear map594

C.2.c Change of basis595

C.2.d Change of basis formula595

C.2.e Case of an orthonormal basis596

D A few proofs597

Tables609

Fourier transforms609

Laplace transforms613

Probability laws616

Further reading617

References621

Portraits627

Sidebars629

Index631