Mathematics for physics and physicists

Mathematics for Physics and Physicists Walter APPEL Translated by Emmanuel Kowalski Princeton University Press Princeton and Oxford Contents A book's apology xviii Index of notation xxii 1 Reminders: convergence of sequences and series 1.1 The problem of limits in physics l.l.a Two paradoxes involving kinetic energy l.l.b Romeo, Juliet, and viscous fluids l.l.c Potential wall in quantum mechanics 1.1.d Semi-infinite filter behaving as waveguide 1.2 Sequences 1.2.a Sequences in a normed vector space 1.2.b Cauchy sequences 1.2.C The fixed point theorem 1.2.d Double sequences 1.2.e Sequential definition of the limit of a function 1.2.f Sequences of functions 1.3 Series 1.3.a Series in a normed vector space 1.3.b Doubly infinite series 1.3.C Convergence of a double series 1.3.d Conditionally convergent series, absolutely convergent series . 1.3.e Series of functions 1.4 Power series, analytic functions 1.4.a Taylor formulas 1.4.b Some numerical illustrations 1.4.c Radius of convergence of a power series 1.4.d Analytic functions 1.5 A quick look at asymptotic and divergent series 1.5.a Asymptotic series 1.5.b Divergent series and asymptotic expansions Exercises Problem Solutions 1 1 1 5 7 9 12 12 13 15 16 17 18 23 23 24 25 26 29 30 31 32 34 35 37 37 38 43 46 47 2 Measure theory and the Lebesgue integral 2.1 The integral according to Mr. Riemann 2.La Riemann sums 2.1.b Limitations of Riemann's definition 2.2 The integral according to Mr. Lebesgue 2.2.a Principle of the method 51 51 51 54 54 55 CONTENTS vi 2.2.b 2.2.C 2.2.d 2.2.e 2.2.f 2.2.g 2.2.h 2.2.i Exercises Solutions Borel subsets Lebesgue measure The Lebesgue a-algebra Negligible sets Lebesgue measure on R" Definition of the Lebesgue integral Functions zero almost everywhere, space L1 And today? Integral calculus 3.1 Integrability in practice 3.1.a Standard functions 3.1.b Comparison theorems 3.2 Exchanging integrals and limits or series 3.3 Integrals with parameters 3.3.a Continuity of functions defined by integrals 3.3.b Differentiating under the integral sign 3.3.C Case of parameters appearing in the integration range . . . . 3.4 Double and multiple integrals 3.5 Change of variables Exercises Solutions Complex Analysis I 4.1 Holomorphic functions 4.1.a Definitions 4.1.b Examples 4.1.C The operators d/dz and d/dz 4.2 Cauchy's theorem 4.2.a Path integration 4.2.b Integrals along a circle 4.2.C Winding number 4.2.d Various forms of Cauchy's theorem 4.2.e Application 4.3 Properties of holomorphic functions 4.3.a The Cauchy formula and applications 4.3.b Maximum modulus principle 4.3.C Other theorems 4.3.d Classification of zero sets of holomorphic functions 4.4 Singularities of a function 4.4.a Classification of singularities 4.4.b Meromorphic functions 4.5 Laurent series 4.5.a Introduction and definition 4.5.b Examples of Laurent series 4.5.C The Residue theorem 4.5.d Practical computations of residues 56 58 59 61 62 62 66 67 68 71 73 73 73 74 75 77 77 78 78 79 81 83 85 87 87 88 90 91 93 93 95 96 96 99 99 99 104 105 106 108 108 110 Ill Ill 113 114 116 CONTENTS Vi i 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 M 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 Complex Analysis II 5.1 Complex logarithm; multivalued functions 5.La The complex logarithms 5.1.b The square root function 5.1.C Multivalued functions, Riemann surfaces 5.2 Harmonic functions 5.2.a Definitions 5.2.b Properties 5.2.c A trick to find / knowing u 5.3 Analytic continuation 5.4 Singularities at infinity 5.5 The saddle point method 5.5.a The general saddle point method 5.5.b The real saddle point method Exercises Solutions 135 135 135 137 137 139 139 140 142 144 146 148 149 152 153 154 Conformal maps 6.1 Conformal maps 6.La Preliminaries , 6.1.b The Riemann mapping theorem 6.1.C Examples of conformal maps 6.1.d The Schwarz-Christoffel transformation 6.2 Applications to potential theory 6.2.a Application to electrostatics 6.2.b Application to hydrodynamics 6.2.C Potential theory, lightning rods, and percolation 6.3 Dirichlet problem and Poisson kernel Exercises Solutions 155 155 155 157 158 161 163 165 167 169 170 174 176 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 viii CONTENTS 7.2.d Other examples Elementary properties. Operations 7.3.a Operations on distributions 7.3.b Derivative of a distribution Dirac and its derivatives 7.4.a The Heaviside distribution 7.4.b Multidimensional Dirac distributions 7.4.C The distribution 8' 7.4.d Composition of 8 with a function . . . .• 7.4.e Charge and current densities Derivation of a discontinuous function 7.5.a Derivation of a function discontinuous at a point 7.5.b Derivative of a function with discontinuity along a surface S? 7.5.c Laplacian of a function discontinuous along a surface Sf • . 7.5.d Application: laplacian of 1/r in 3-space Convolution 7.6.a The tensor product of two functions 7.6.b The tensor product of distributions 7.6.c Convolution of two functions 7.6.d "Fuzzy" measurement 7.6.e Convolution of distributions 7.6.f Applications 7.6.g The Poisson equation Physical interpretation of convolution operators . . Discrete convolution 187 188 188 191 193 193 194 196 198 199 201 201 204 206 207 209 209 209 211 213 214 215 216 217 220 Distributions II 8.1 Cauchy principal value 8.La Definition 8.1.b Application to the computation of certain integrals 8.1.C Feynman's notation :• 8.1.d Kramers-Kronig relations 8.1.e A few equations in the sense of distributions 8.2 Topology in &' 8.2.a Weak convergence in ty 8.2.b Sequences of functions converging to 8 8.2.C Convergence in Si1 and convergence in the sense of functions 8.2.d Regularization of a distribution 8.2.e Continuity of convolution 8.3 Convolution algebras 8.4 Solving a differential equation with initial conditions 8.4.a First order equations 8.4.b The case of the harmonic oscillator 8.4.c Other equations of physical origin Exercises Problem Solutions 223 223 223 224 225 227 229 230 230 231 234 234 235 236 238 238 239 240 241 244 245 7.3 7.4 7.5 7.6 7.7 7.8 CONTENTS 9 ix Hilbert spaces; Fourier series 9.1 Insufficiency of vector spaces 9.2 Pre-Hilbert spaces 9.2.a The finite-dimensional case 9.2.b Projection on a finite-dimensional subspace 9.2.C Bessel inequality 9.3 Hilbert spaces 9.3.a Hilbert basis 9.3.b The I2 space 9.3.c The space L2 [0,a] 9.3.d The L 2 (R) space 9.4 Fourier series expansion 9.4.a Fourier coefficients of a function 9.4.b Mean-square convergence 9.4.C Fourier series of a function / e L1 [0,a] 9.4.d Pointwise convergence of the Fourier series 9.4.e Uniform convergence of the Fourier series 9.4.f The Gibbs phenomenon Exercises Problem Solutions 10 Fourier transform of functions 10.1 Fourier transform of a function in L1 lO.l.a Definition lO.l.b Examples lO.l.c The L1 space 10.1.d Elementary properties lO.l.e Inversion lO.l.f Extension of the inversion formula 10.2 Properties of the Fourier transform . . . . ' , 10.2.a Transpose and translates 10.2.b Dilation 10.2.C Derivation 10.2.d Rapidly decaying functions 10.3 Fourier transform of a function in L2 10.3.a The space S? 10.3.b The Fourier transform in L2 10.4 Fourier transform and convolution 10.4.a Convolution formula 10.4.b Cases of the convolution formula Exercises Solutions 11 Fourier transform of distributions 11.1 Definition and properties ll.l.a Tempered distributions ll.l.b Fourier transform of tempered distributions ll.l.c Examples 249 249 251 254 254 256 256 257 261 262 263 264 264 265 266 267 269 270 270 271 272 ; 277 277 278 279 279 280 282 284 285 285 . . . . 286 286 288 288 289 290 292 292 293 295 296 299 299 300 301 303 CONTENTS ll.l.d Higher-dimensional Fourier transforms ll.l.e Inversion formula 11.2 The Dirac comb 11.2.a Definition and properties 11.2.b Fourier transform of a periodic function 11.2.C Poisson summation formula 11.2.d Application to the computation of series 11.3 The Gibbs phenomenon 11.4 Application to physical optics 11.4.a Link between diaphragm and diffraction figure 11.4.b Diaphragm made of infinitely many infinitely narrow slits 11.4.C Finite number of infinitely narrow slits 11.4.d Finitely many slits with finite width 11.4.e Circular lens 11.5 Limitations of Fourier analysis and wavelets Exercises Problem Solutions 305 306 307 307 308 309 310 311 314 314 . 315 316 318 320 321 324 325 326 12 The Laplace transform 12.1 Definition and integrability 12.1.a Definition 12.1.b Integrability 12.1.C Properties of the Laplace transform 12.2 Inversion 12.3 Elementary properties and examples of Laplace transforms 12.3.a Translation 12.3.b Convolution 12.3.C Differentiation and integration 12.3.d Examples 12.4 Laplace transform of distributions . . . : 12.4.a Definition 12.4.b Properties 12.4.C Examples 12.4.d The z-transform 12.4.e Relation between Laplace and Fourier transforms 12.5 Physical applications, the Cauchy problem 12.5.a Importance of the Cauchy problem 12.5.b A simple example 12.5.c Dynamics of the electromagnetic field without sources . . . . Exercises Solutions 331 331 332 333 336 336 338 338 339 339 341 342 342 342 344 344 345 346 346 347 348 351 352 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 CONTENTS X i 13.5.a Definition 13.5.b Properties 13.5.C Intercorrelation 13.6 Finite power functions 13.6.a Definitions 13.6.b Autocorrelation 13.7 Application to optics: the Wiener-Khintchine theorem Exercises Solutions 14 Bras, kets, and all that sort of thing 14.1 Reminders about finite dimension 14.La Scalar product and representation theorem 14.1.b Adjoint 14.1.C Symmetric and hermitian endomorphisms 14.2 Kets and bras 14.2.a Kets |</>) e H 14.2.b Bras {<p\eH' 14.2.C Generalized bras 14.2.d Generalized kets 14.2.e 14.2.f Id = ! ; > , , ) Generalized basis fo,| 14.3 Linear operators 14.3.a Operators 14.3.b Adjoint 14.3.c Bounded operators, closed operators, closable operators . . . 14.3.d Discrete and continuous spectra 14.4 Hermitian operators; self-adjoint operators 14.4.a Definitions 14.4.b Eigenvectors 14.4.c Generalized eigenvectors ; 14.4.d "Matrix" representation 14.4.e Summary of properties of the operators P and X Exercises Solutions 15 Green functions 15.1 Generalities about Green functions 15.2 A pedagogical example: the harmonic oscillator 15.2.a Using the Laplace transform 15.2.b Using the Fourier transform 15.3 Electromagnetism and the d'Alembertian operator 15.3.a Computation of the advanced and retarded Green functions 15.3.b Retarded potentials 15.3.C Covariant expression of advanced and retarded Green functions 15.3.d Radiation 15.4 The heat equation 15.4.a One-dimensional case 15.4.b Three-dimensional case 368 368 369 370 370 370 371 375 376 377 377 377 378 379 379 379 380 382 383 384 385 387 387 389 390 391 393 394 396 397 398 401 403 404 407 407 409 410 410 414 414 418 421 421 422 423 426 x iv CONTENTS 20.10.b Application: Buffon's needle 20.11 Independance, correlation, causality 21 Convergence of random variables: central limit theorem 21.1 Various types of convergence 21.2 The law of large numbers 21.3 Central limit theorem Exercises Problems Solutions 549 550 553 553 555 556 560 563 564 Appendices A Reminders concerning topology and normed vector spaces A.I Topology, topological spaces A.2 Normed vector spaces A.2.a Norms, seminorms A.2.b Balls and topology associated to the distance A.2.C Comparison of sequences A.2.d Bolzano-Weierstrass theorems A.2.e Comparison of norms : A.2.f Norm of a linear map Exercise Solution 573 573 577 577 578 580 581 581 583 583 584 B Elementary reminders of differential calculus B.I Differential of a real-valued function B.l.a Functions of one real variable . l B.l.b Differential of a function / : M." -» R B.l.c Tensor notation B.2 Differential of map with values in W B.3 Lagrange multipliers Solution 585 585 585 586 587 587 588 591 C Matrices C.I Duality C.2 Application to matrix representation C.2.a Matrix representing a family of vectors C.2.b Matrix of a linear map C.2.C Change of basis C.2.d Change of basis formula C.2.e Case of an orthonormal basis 593 593 594 594 594 595 595 596 D A few proofs 597 CONTENTS XV Tables Fourier transforms 609 Laplace transforms 613 Probability laws 616 Further reading 617 References 621 Portraits 627 Sidebars 629 Index 631