Mathematics for Physics and Physicists

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1 Mathematics for Physics and Physicists Walter APPEL Translated by Emmanuel Kowalski Princeton University Press Princeton and Oxford

2 Contents A book's apology Index of notation xviii xxii 1 Reminders: convergence of sequences and series The problem of limits in physics 1 l.l.a Two paradoxes involving kinetic energy 1 l.l.b Romeo, Juliet, and viscous fluids 5 l.l.c Potential wall in quantum mechanics d Semi-infinite filter behaving as waveguide Sequences a Sequences in a normed vector space b Cauchy sequences C The fixed point theorem d Double sequences e Sequential definition of the limit of a function f Sequences of functions Series a Series in a normed vector space b Doubly infinite series C Convergence of a double series d Conditionally convergent series, absolutely convergent series e Series of functions Power series, analytic functions a Taylor formulas b Some numerical illustrations c Radius of convergence of a power series d Analytic functions A quick look at asymptotic and divergent series a Asymptotic series b Divergent series and asymptotic expansions 38 Exercises 43 Problem 46 Solutions 47 2 Measure theory and the Lebesgue integral The integral according to Mr. Riemann 51 2.La Riemann sums b Limitations of Riemann's definition The integral according to Mr. Lebesgue a Principle of the method 55

3 v i CONTENTS 2.2.b Borel subsets C Lebesgue measure d The Lebesgue a-algebra e Negligible sets f Lebesgue measure on R" g Definition of the Lebesgue integral h Functions zero almost everywhere, space L i And today? 67 Exercises 68 Solutions 71 Integral calculus Integrability in practice a Standard functions b Comparison theorems Exchanging integrals and limits or series Integrals with parameters a Continuity of functions defined by integrals b Differentiating under the integral sign C Case of parameters appearing in the integration range Double and multiple integrals Change of variables 81 Exercises 83 Solutions 85 Complex Analysis I Holomorphic functions a Definitions b Examples C The operators d/dz and d/dz Cauchy's theorem a Path integration b Integrals along a circle C Winding number d Various forms of Cauchy's theorem e Application Properties of holomorphic functions a The Cauchy formula and applications b Maximum modulus principle C Other theorems d Classification of zero sets of holomorphic functions Singularities of a function a Classification of singularities b Meromorphic functions Laurent series Ill 4.5.a Introduction and definition Ill 4.5.b Examples of Laurent series C The Residue theorem d Practical computations of residues 116

4 CONTENTS V i i 4.6 Applications to the computation of horrifying integrals or ghastly sums a Jordan's lemmas b Integrals on M of a rational function C Fourier integrals d Integral on the unit circle of a rational function e Computation of infinite sums 122 Exercises 125 Problem 128 Solutions 129 Complex Analysis II Complex logarithm; multivalued functions La The complex logarithms b The square root function C Multivalued functions, Riemann surfaces Harmonic functions a Definitions b Properties c A trick to find / knowing u Analytic continuation Singularities at infinity The saddle point method a The general saddle point method b The real saddle point method 152 Exercises 153 Solutions 154 Conformal maps Conformal maps La Preliminaries, b The Riemann mapping theorem C Examples of conformal maps d The Schwarz-Christoffel transformation Applications to potential theory a Application to electrostatics b Application to hydrodynamics C Potential theory, lightning rods, and percolation Dirichlet problem and Poisson kernel 170 Exercises 174 Solutions 176 Distributions I Physical approach a The problem of distribution of charge b The problem of momentum and forces during an elastic shock Definitions and examples of distributions a Regular distributions b Singular distributions C Support of a distribution 187

5 viii CONTENTS 7.2.d Other examples Elementary properties. Operations a Operations on distributions b Derivative of a distribution Dirac and its derivatives a The Heaviside distribution b Multidimensional Dirac distributions C The distribution 8' d Composition of 8 with a function e Charge and current densities Derivation of a discontinuous function a Derivation of a function discontinuous at a point b Derivative of a function with discontinuity along a surface S? c Laplacian of a function discontinuous along a surface Sf d Application: laplacian of 1/r in 3-space Convolution a The tensor product of two functions b The tensor product of distributions c Convolution of two functions d "Fuzzy" measurement e Convolution of distributions f Applications g The Poisson equation Physical interpretation of convolution operators Discrete convolution 220 Distributions II Cauchy principal value La Definition b Application to the computation of certain integrals C Feynman's notation : d Kramers-Kronig relations e A few equations in the sense of distributions Topology in &' a Weak convergence in ty b Sequences of functions converging to C Convergence in Si 1 and convergence in the sense of functions d Regularization of a distribution e Continuity of convolution Convolution algebras Solving a differential equation with initial conditions a First order equations b The case of the harmonic oscillator c Other equations of physical origin 240 Exercises 241 Problem 244 Solutions 245

6 CONTENTS ix 9 Hilbert spaces; Fourier series Insufficiency of vector spaces Pre-Hilbert spaces a The finite-dimensional case b Projection on a finite-dimensional subspace C Bessel inequality Hilbert spaces a Hilbert basis b The I 2 space c The space L 2 [0,a] d The L 2 (R) space Fourier series expansion a Fourier coefficients of a function b Mean-square convergence C Fourier series of a function / e L 1 [0,a] d Pointwise convergence of the Fourier series e Uniform convergence of the Fourier series f The Gibbs phenomenon 270 Exercises 270 Problem 271 Solutions Fourier transform of functions Fourier transform of a function in L lo.l.a Definition 278 lo.l.b Examples 279 lo.l.c The L 1 space d Elementary properties 280 lo.l.e Inversion 282 lo.l.f Extension of the inversion formula Properties of the Fourier transform...', a Transpose and translates b Dilation ; C Derivation d Rapidly decaying functions Fourier transform of a function in L a The space S? b The Fourier transform in L Fourier transform and convolution a Convolution formula b Cases of the convolution formula 293 Exercises 295 Solutions Fourier transform of distributions Definition and properties 299 ll.l.a Tempered distributions 300 ll.l.b Fourier transform of tempered distributions 301 ll.l.c Examples 303

7 CONTENTS ll.l.d Higher-dimensional Fourier transforms 305 ll.l.e Inversion formula The Dirac comb a Definition and properties b Fourier transform of a periodic function C Poisson summation formula d Application to the computation of series The Gibbs phenomenon Application to physical optics a Link between diaphragm and diffraction figure b Diaphragm made of infinitely many infinitely narrow slits C Finite number of infinitely narrow slits d Finitely many slits with finite width e Circular lens Limitations of Fourier analysis and wavelets 321 Exercises 324 Problem 325 Solutions The Laplace transform Definition and integrability a Definition b Integrability C Properties of the Laplace transform Inversion Elementary properties and examples of Laplace transforms a Translation b Convolution C Differentiation and integration d Examples Laplace transform of distributions... : a Definition b Properties C Examples d The z-transform e Relation between Laplace and Fourier transforms Physical applications, the Cauchy problem a Importance of the Cauchy problem b A simple example c Dynamics of the electromagnetic field without sources Exercises 351 Solutions Physical applications of the Fourier transform Justification of sinusoidal regime analysis Fourier transform of vector fields: longitudinal and transverse fields Heisenberg uncertainty relations Analytic signals Autocorrelation of a finite energy function 368

8 CONTENTS X i 13.5.a Definition b Properties C Intercorrelation Finite power functions a Definitions b Autocorrelation Application to optics: the Wiener-Khintchine theorem 371 Exercises 375 Solutions Bras, kets, and all that sort of thing Reminders about finite dimension La Scalar product and representation theorem b Adjoint C Symmetric and hermitian endomorphisms Kets and bras a Kets </>) e H b Bras {<p\eh' C Generalized bras d Generalized kets e Id =!;>,,) fo, f Generalized basis Linear operators a Operators b Adjoint c Bounded operators, closed operators, closable operators d Discrete and continuous spectra Hermitian operators; self-adjoint operators a Definitions b Eigenvectors c Generalized eigenvectors ; d "Matrix" representation e Summary of properties of the operators P and X 401 Exercises 403 Solutions Green functions Generalities about Green functions A pedagogical example: the harmonic oscillator a Using the Laplace transform b Using the Fourier transform Electromagnetism and the d'alembertian operator a Computation of the advanced and retarded Green functions b Retarded potentials C Covariant expression of advanced and retarded Green functions d Radiation The heat equation a One-dimensional case b Three-dimensional case 426

9 x i v CONTENTS b Application: Buffon's needle Independance, correlation, causality Convergence of random variables: central limit theorem Various types of convergence The law of large numbers Central limit theorem 556 Exercises 560 Problems 563 Solutions 564 Appendices A Reminders concerning topology and normed vector spaces 573 A.I 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.I Differential of a real-valued function 585 B.l.a Functions of one real variable. l 585 B.l.b Differential of a function / : M." -» R 586 B.l.c Tensor notation 587 B.2 Differential of map with values in W 587 B.3 Lagrange multipliers 588 Solution 591 C Matrices 593 C.I 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

10 CONTENTS X V Tables Fourier transforms 609 Laplace transforms 613 Probability laws 616 Further reading 617 References 621 Portraits 627 Sidebars 629 Index 631

A book s apology. [This introduction was intented for a French reader. Its first page needs rewriting for a wider audience.]

A book s apology. [This introduction was intented for a French reader. Its first page needs rewriting for a wider audience.] A book s apology [This introduction was intented for a French reader. Its first page needs rewriting for a wider audience.] W hy should a physicist study mathematics? There is a fairly fashionable current