数学物理 英文PDF电子书下载

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