Mathematics for Physics and Physicists > 물리학

A book\'s apology xviii
Index of notation xxii
Reminders: convergence of sequences and series 1
The problem of limits in physics 1
Two paradoxes involving kinetic energy 1
Romeo, Juliet, and viscous fluids 5
Potential wall in quantum mechanics 7
Semi-infinite filter behaving as waveguide 9
Sequences 12
Sequences in a normed vector space 12
Cauchy sequences 13
The fixed point theorem 15
Double sequences 16
Sequential definition of the limit of a function 17
Sequences of functions 18
Series 23
Series in a normed vector space 23
Doubly infinite series 24
Convergence of a double series 25
Conditionally convergent series, absolutely convergent series 26
Series of functions 29
Power series, analytic functions 30
Taylor formulas 31
Some numerical illustrations 32
Radius of convergence of a power series 34
Analytic functions 35
A quick look at asymptotic and divergent series 37
Asymptotic series 37
Divergent series and asymptotic expansions 38
Exercises 43
Problem 46
Solutions 47
Measure theory and the Lebesgue integral 51
The integral according to Mr. Riemann 51
Riemann sums 51
Limitations of Riemann\'s definition 54
The integral according to Mr. Lebesgue 54
Principle of the method 55
Borel subsets 56
Lebesgue measure 58
The Lebesgue [sigma]-algebra 59
Negligible sets 61
Lebesgue measure on R[superscript n] 62
Definition of the Lebesgue integral 62
Functions zero almost everywhere, space L[superscript 1] 66
And today? 67
Exercises 68
Solutions 71
Integral calculus 73
Integrability in practice 73
Standard functions 73
Comparison theorems 74
Exchanging integrals and limits or series 75
Integrals with parameters 77
Continuity of functions defined by integrals 77
Differentiating under the integral sign 78
Case of parameters appearing in the integration range 78
Double and multiple integrals 79
Change of variables 81
Exercises 83
Solutions 85
Complex Analysis I 87
Holomorphic functions 87
Definitions 88
Examples 90
The operators [part]/[part]z and [part]/[part]z 91
Cauchy\'s theorem 93
Path integration 93
Integrals along a circle 95
Winding number 96
Various forms of Cauchy\'s theorem 96
Application 99
Properties of holomorphic functions 99
The Cauchy formula and applications 99
Maximum modulus principle 104
Other theorems 105
Classification of zero sets of holomorphic functions 106
Singularities of a function 108
Classification of singularities 108
Meromorphic functions 110
Laurent series 111
Introduction and definition 111
Examples of Laurent series 113
The Residue theorem 114
Practical computations of residues 116
Applications to the computation of horrifying integrals or ghastly sums 117
Jordan\'s lemmas 117
Integrals on R of a rational function 118
Fourier integrals 120
Integral on the unit circle of a rational function 121
Computation of infinite sums 122
Exercises 125
Problem 128
Solutions 129
Complex Analysis II 135
Complex logarithm; multivalued functions 135
The complex logarithms 135
The square root function 137
Multivalued functions, Riemann surfaces 137
Harmonic functions 139
Definitions 139
Properties 140
A trick to find f knowing u 142
Analytic continuation 144
Singularities at infinity 146
The saddle point method 148
The general saddle point method 149
The real saddle point method 152
Exercises 153
Solutions 154
Conformal maps 155
Conformal maps 155
Preliminaries 155
The Riemann mapping theorem 157
Examples of conformal maps 158
The Schwarz-Christoffel transformation 161
Applications to potential theory 163
Application to electrostatics 165
Application to hydrodynamics 167
Potential theory, lightning rods, and percolation 169
Dirichlet problem and Poisson kernel 170
Exercises 174
Solutions 176
Distributions I 179
Physical approach 179
The problem of distribution of charge 179
The problem of momentum and forces during an elastic shock 181
Definitions and examples of distributions 182
Regular distributions 184
Singular distributions 185
Support of a distribution 187
Other examples 187
Elementary properties. Operations 188
Operations on distributions 188
Derivative of a distribution 191
Dirac and its derivatives 193
The Heaviside distribution 193
Multidimensionai Dirac distributions 194
The distribution [delta]\' 196
Composition of [delta] with a function 198
Charge and current densities 199
Derivation of a discontinuous function 201
Derivation of a function discontinuous at a point 201
Derivative of a function with discontinuity along a surface L 204
Laplacian of a function discontinuous along a surface L 206
Application: laplacian of 1/r in 3-space 207
Convolution 209
The tensor product of two functions 209
The tensor product of distributions 209
Convolution of two functions 211
\"Fuzzy\" measurement 213
Convolution of distributions 214
Applications 215
The Poisson equation 216
Physical interpretation of convolution operators 217
Discrete convolution 220
Distributions II 223
Cauchy principal value 223
Definition 223
Application to the computation of certain integrals 224
Feynman\'s notation 225
Kramers-Kronig relations 227
A few equations in the sense of distributions 229
Topology D\' 230
Weak convergence in D\' 230
Sequences of functions converging to [delta] 231
Convergence in D\' and convergence in the sense of functions 234
Regularization of a distribution 234
Continuity of convolution 235
Convolution algebras 236
Solving a differential equation with initial conditions 238
First order equations 238
The case of the harmonic oscillator 239
Other equations of physical origin 240
Exercises 241
Problem 244
Solutions 245
Hilbert spaces; Fourier series 249
Insufficiency of vector spaces 249
Pre-Hilbert spaces 251
The finite-dimensional case 254
Projection on a finite-dimensional subspace 254
Bessel inequality 256
Hilbert spaces 256
Hilbert basis 257
The [ell superscript 2] space 261
The space L[superscript 2] [0,a] 262
The L[superscript 2](R) space 263
Fourier series expansion 264
Fourier coefficients of a function 264
Mean-square convergence 265
Fourier series of a function f [Element] L[superscript 1] [0,a] 266
Pointwise convergence of the Fourier series 267
Uniform convergence of the Fourier series 269
The Gibbs phenomenon 270
Exercises 270
Problem 271
Solutions 272
Fourier transform of functions 277
Fourier transform of a function in L[superscript 1] 277
Definition 278
Examples 279
The L[superscript 1] space 279
Elementary properties 280
Inversion 282
Extension of the inversion formula 284
Properties of the Fourier transform 285
Transpose and translates 285
Dilation 286
Derivation 286
Rapidly decaying functions 288
Fourier transform of a function in L[superscript 2] 288
The space L 289
The Fourier transform in L[superscript 2] 290
Fourier transform and convolution 292
Convolution formula 292
Cases of the convolution formula 293
Exercises 295
Solutions 296
Fourier transform of distributions 299
Definition and properties 299
Tempered distributions 300
Fourier transform of tempered distributions 301
Examples 303
Higher-dimensional Fourier transforms 305
Inversion formula 306
The Dirac comb 307
Definition and properties 307
Fourier transform of a periodic function 308
Poisson summation formula 309
Application to the computation of series 310
The Gibbs phenomenon 311
Application to physical optics 314
Link between diaphragm and diffraction figure 314
Diaphragm made of infinitely many infinitely narrow slits 315
Finite number of infinitely narrow slits 316
Finitely many slits with finite width 318
Circular lens 320
Limitations of Fourier analysis and wavelets 321
Exercises 324
Problem 325
Solutions 326
The Laplace transform 331
Definition and integrability 331
Definition 332
Integrability 333
Properties of the Laplace transform 336
Inversion 336
Elementary properties and examples of Laplace transforms 338
Translation 338
Convolution 339
Differentiation and integration 339
Examples 341
Laplace transform of distributions 342
Definition 342
Properties 342
Examples 344
The z-transform 344
Relation between Laplace and Fourier transforms 345
Physical applications, the Cauchy problem 346
Importance of the Cauchy problem 346
A simple example 347
Dynamics of the electromagnetic field without sources 348
Exercises 351
Solutions 352
Physical applications of the Fourier transform 355
Justification of sinusoidal regime analysis 355
Fourier transform of vector fields: longitudinal and transverse fields 358
Heisenberg uncertainty relations 359
Analytic signals 365
Autocorrelation of a finite energy function 368
Definition 368
Properties 368
Intercorrelation 369
Finite power functions 370
Definitions 370
Autocorrelation 370
Application to optics: the Wiener-Khintchine theorem 371
Exercises 375
Solutions 376
Bras, kets, and all that sort of thing 377
Reminders about finite dimension 377
Scalar product and representation theorem 377
Adjoint 378
Symmetric and hermitian endomorphisms 379
Kets and bras 379
Kets [Characters not reproducible] [Element] H 379
Bras [Characters not reproducible] [Element] H\' 380
Generalized bras 382
Generalized kets 383
Id = [Sigma subscript n] | [phi subscript n]> <[phi subscript n]| 384
Generalized basis 385
Linear operators 387
Operators 387
Adjoint 389
Bounded operators, closed operators, closable operators 390
Discrete and continuous spectra 391
Hermitian operators; self-adjoint operators 393
Definitions 394
Eigenvectors 396
Generalized eigenvectors 397
\"Matrix\" representation 398
Summary of properties of the operators P and X 401
Exercises 403
Solutions 404
Green functions 407
Generalities about Green functions 407
A pedagogical example: the harmonic oscillator 409
Using the Laplace transform 410
Using the Fourier transform 410
Electromagnetism and the d\'Alembertian operator 414
Computation of the advanced and retarded Green functions 414
Retarded potentials 418
Covariant expression of advanced and retarded Green functions 421
Radiation 421
The heat equation 422
One-dimensional case 423
Three-dimensional case 426
Quantum mechanics 427
Klein-Gordon equation 429
Exercises 432
Tensors 433
Tensors in affine space 433
Vectors 433
Einstein convention 435
Linear forms 436
Linear maps 438
Lorentz transformations 439
Tensor product of vector spaces: tensors 439
Existence of the tensor product of two vector spaces 439
Tensor product of linear forms: tensors of type [Characters not reproducible] 441
Tensor product of vectors: tensors of type [Characters not reproducible] 443
Tensor product of a vector and a linear form: linear maps or [Characters not reproducible]-tensors 444
Tensors of type [Characters not reproducible] 446
The metric, or, how to raise and lower indices 447
Metric and pseudo-metric 447
Natural duality by means of the metric 449
Gymnastics: raising and lowering indices 450
Operations on tensors 453
Change of coordinates 455
Curvilinear coordinates 455
Basis vectors 456
Transformation of physical quantities 458
Transformation of linear forms 459
Transformation of an arbitrary tensor field 460
Conclusion 461
Solutions 462
Differential forms 463
Exterior algebra 463
1-forms 463
Exterior 2-forms 464
Exterior k-forms 465
Exterior product 467
Differential forms on a vector space 469
Definition 469
Exterior derivative 470
Integration of differential forms 471
Poincare\'s theorem 474
Relations with vector calculus: gradient, divergence, curl 476
Differential forms in dimension 3 476
Existence of the scalar electrostatic potential 477
Existence of the vector potential 479
Magnetic monopoles 480
Electromagnetism in the language of differential forms 480
Problem 484
Solution 485
Groups and group representations 489
Groups 489
Linear representations of groups 491
Vectors and the group SO(3) 492
The group SU(2) and spinors 497
Spin and Riemann sphere 503
Exercises 505
Introduction to probability theory 509
Introduction 510
Basic definitions 512
Poincare formula 516
Conditional probability 517
Independent events 519
Random variables 521
Random variables and probability distributions 521
Distribution function and probability density 524
Discrete random variables 526
(Absolutely) continuous random variables 526
Expectation and variance 527
Case of a discrete r.v. 527
Case of a continuous r.v. 528
An example: the Poisson distribution 530
Particles in a confined gas 530
Radioactive decay 531
Moments of a random variable 532
Random vectors 534
Pair of random variables 534
Independent random variables 537
Random vectors 538
Image measures 539
Case of a single random variable 539
Case of a random vector 540
Expectation and characteristic function 540
Expectation of a function of random variables 540
Moments, variance 541
Characteristic function 541
Generating function 543
Sum and product of random variables 543
Sum of random variables 543
Product of random variables 546
Example: Poisson distribution 547
Bienayme-Tchebychev inequality 547\\
Statement 547
Application: Buffon\'s needle 549
Independance, correlation, causality 550
Convergence of random variables: central limit theorem 553
Various types of convergence 553
The law of large numbers 555
Central limit theorem 556
Exercises 560
Problems 563
Solutions 564
Appendices
Reminders concerning topology and normed vector spaces 573
Topology, topological spaces 573
Normed vector spaces 577
Norms, seminorms 577
Balls and topology associated to the distance 578
Comparison of sequences 580
Bolzano-Weierstrass theorems 581
Comparison of norms 581
Norm of a linear map 583
Exercise 583
Solution 584
Elementary reminders of differential calculus 585
Differential of a real-valued function 585
Functions of one real variable 585
Differential of a function f : R[superscript n] [right arrow] R 586
Tensor notation 587
Differential of map with values in R[superscript p] 587
Lagrange multipliers 588
Solution 591
Matrices 593
Duality 593
Application to matrix representation 594
Matrix representing a family of vectors 594
Matrix of a linear map 594
Change of basis 595
Change of basis formula 595
Case of an orthonormal basis 596
A few proofs 597
Tables
Fourier transforms 609
Laplace transforms 613
Probability laws 616
Further reading 617
References 621
Portraits 627
Sidebars 629
Index 631