Math - Appel

8/2/2019 Math - Appel

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Mathematics for Physicsand Physicists


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Mathematics for Physics

and Physicists

Walter Appel

Translated by Emmanuel Kowalski

Princeton University PressPrinceton and Oxford


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Copyright c 2007 by Princeton University Press

Published by Princeton University Press, 41 William Street,Princeton, New Jersey 08540In the United Kingdom: Princeton University Press, 3 Market Place,Woodstock, Oxfordshire OX20 1SY

Originally published in French under the title Mathmatiques pour la physique... et lesphysiciens!, copyright c 2001 by H&K Editions, Paris.

Translated from the French by Emmanuel Kowalski

Ouvrage publi avec le concours du Ministre franais charg de la culture Centrenational du livre.

[This book was published with the support of the French Ministry of Culture Centre national du livre.]

All Rights ReservedLibrary of Congress Control Number: 2006935871

ISBN-13: 978-0-691-13102-3

ISBN-10: 0-691-13102-3British Library Cataloging-in-Publication Data is available

Printed on acid-free paper.

pup.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Formatted with LA

TEX 2 under a Linux environment.


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Contents

A books apology xviii

Index of notation xxii

1 Reminders: convergence of sequences and series 11.1 The problem of limits in physics . . . . . . . . . . . . . . . . . . . . . . 1

1.1.a Two paradoxes involving kinetic energy . . . . . . . . . . . . . 11.1.b Romeo, Juliet, and viscous fluids . . . . . . . . . . . . . . . . . 51.1.c Potential wall in quantum mechanics . . . . . . . . . . . . . . 71.1.d Semi-infinite filter behaving as waveguide . . . . . . . . . . . . 9

1.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.a Sequences in a normed vector space . . . . . . . . . . . . . . . 121.2.b Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.c The fixed point theorem . . . . . . . . . . . . . . . . . . . . . . 151.2.d Double sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.e Sequential definition of the limit of a function . . . . . . . . 171.2.f Sequences of functions . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.a Series in a normed vector space . . . . . . . . . . . . . . . . . 231.3.b Doubly infinite series . . . . . . . . . . . . . . . . . . . . . . . 241.3.c Convergence of a double series . . . . . . . . . . . . . . . . . . 251.3.d Conditionally convergent series, absolutely convergent series . 261.3.e Series of functions . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Power series, analytic functions . . . . . . . . . . . . . . . . . . . . . . . 301.4.a Taylor formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4.b Some numerical illustrations . . . . . . . . . . . . . . . . . . . 321.4.c Radius of convergence of a power series . . . . . . . . . . . . . 341.4.d Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 A quick look at asymptotic and divergent series . . . . . . . . . . . . . 37

1.5.a Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . 371.5.b Divergent series and asymptotic expansions . . . . . . . . . . 38Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Measure theory and the Lebesgue integral 512.1 The integral according to Mr. Riemann . . . . . . . . . . . . . . . . . . 51

2.1.a Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.b Limitations of Riemanns definition . . . . . . . . . . . . . . . 54

2.2 The integral according to Mr. Lebesgue . . . . . . . . . . . . . . . . . . 542.2.a Principle of the method . . . . . . . . . . . . . . . . . . . . . . 55


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2.2.b Borel subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2.c Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.d The Lebesgue -algebra . . . . . . . . . . . . . . . . . . . . . . 60

2.2.e Negligible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2.f Lebesgue measure on n . . . . . . . . . . . . . . . . . . . . . 622.2.g Definition of the Lebesgue integral . . . . . . . . . . . . . . . 632.2.h Functions zero almost everywhere, space L1 . . . . . . . . . . 662.2.i And today? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Integral calculus 733.1 Integrability in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1.a Standard functions . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1.b Comparison theorems . . . . . . . . . . . . . . . . . . . . . . . 743.2 Exchanging integrals and limits or series . . . . . . . . . . . . . . . . . 753.3 Integrals with parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.a Continuity of functions defined by integrals . . . . . . . . . . 773.3.b Differentiating under the integral sign . . . . . . . . . . . . . 783.3.c Case of parameters appearing in the integration range . . . . 78

3.4 Double and multiple integrals . . . . . . . . . . . . . . . . . . . . . . . 793.5 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


4 Complex Analysis I 87

4.1 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1.a Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.b Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.1.c The operators /z and /z . . . . . . . . . . . . . . . . . . 91

4.2 Cauchys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.a Path integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.b Integrals along a circle . . . . . . . . . . . . . . . . . . . . . . . 954.2.c Winding number . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2.d Various forms of Cauchys theorem . . . . . . . . . . . . . . . 964.2.e Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Properties of holomorphic functions . . . . . . . . . . . . . . . . . . . 994.3.a The Cauchy formula and applications . . . . . . . . . . . . . . 994.3.b Maximum modulus principle . . . . . . . . . . . . . . . . . . . 1044.3.c Other theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.d Classification of zero sets of holomorphic functions . . . . . 106

4.4 Singularities of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.4.a Classification of singularities . . . . . . . . . . . . . . . . . . . 1084.4.b Meromorphic functions . . . . . . . . . . . . . . . . . . . . . . 110

4.5 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.5.a Introduction and definition . . . . . . . . . . . . . . . . . . . 1114.5.b Examples of Laurent series . . . . . . . . . . . . . . . . . . . . 1134.5.c The Residue theorem . . . . . . . . . . . . . . . . . . . . . . . . 1144.5.d Practical computations of residues . . . . . . . . . . . . . . . . 116


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4.6 Applications to the computation of horrifying integrals or ghastly sums 1174.6.a Jordans lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.6.b Integrals on of a rational function . . . . . . . . . . . . . . 1184.6.c Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6.d Integral on the unit circle of a rational function . . . . . . . 1214.6.e Computation of infinite sums . . . . . . . . . . . . . . . . . . 122

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5 Complex Analysis II 1355.1 Complex logarithm; multivalued functions . . . . . . . . . . . . . . . . 135

5.1.a The complex logarithms . . . . . . . . . . . . . . . . . . . . . . 1355.1.b The square root function . . . . . . . . . . . . . . . . . . . . . 137

5.1.c Multivalued functions, Riemann surfaces . . . . . . . . . . . . 1375.2 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2.a Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2.b Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2.c A trick to find f knowing u . . . . . . . . . . . . . . . . . . . 142

5.3 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.4 Singularities at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.5 The saddle point method . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5.a The general saddle point method . . . . . . . . . . . . . . . . 1495.5.b The real saddle point method . . . . . . . . . . . . . . . . . . . 152


6 Conformal maps 1556.1 Conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.1.a Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.1.b The Riemann mapping theorem . . . . . . . . . . . . . . . . . 1576.1.c Examples of conformal maps . . . . . . . . . . . . . . . . . . . 1586.1.d The Schwarz-Christoffel transformation . . . . . . . . . . . . . 161

6.2 Applications to potential theory . . . . . . . . . . . . . . . . . . . . . . 1636.2.a Application to electrostatics . . . . . . . . . . . . . . . . . . . . 1656.2.b Application to hydrodynamics . . . . . . . . . . . . . . . . . . 1676.2.c Potential theory, lightning rods, and percolation . . . . . . . 169

6.3 Dirichlet problem and Poisson kernel . . . . . . . . . . . . . . . . . . . 170Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7 Distributions I 1797.1 Physical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.1.a The problem of distribution of charge . . . . . . . . . . . . . 1797.1.b The problem of momentum and forces during an elastic shock 181

7.2 Definitions and examples of distributions . . . . . . . . . . . . . . . . 1827.2.a Regular distributions . . . . . . . . . . . . . . . . . . . . . . . . 1847.2.b Singular distributions . . . . . . . . . . . . . . . . . . . . . . . 1857.2.c Support of a distribution . . . . . . . . . . . . . . . . . . . . . 187


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7.2.d Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1877.3 Elementary properties. Operations . . . . . . . . . . . . . . . . . . . . . 188

7.3.a Operations on distributions . . . . . . . . . . . . . . . . . . . 188

7.3.b Derivative of a distribution . . . . . . . . . . . . . . . . . . . . 1917.4 Dirac and its derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.4.a The Heaviside distribution . . . . . . . . . . . . . . . . . . . . 1937.4.b Multidimensional Dirac distributions . . . . . . . . . . . . . . 1947.4.c The distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1967.4.d Composition of with a function . . . . . . . . . . . . . . . 1987.4.e Charge and current densities . . . . . . . . . . . . . . . . . . . 199

7.5 Derivation of a discontinuous function . . . . . . . . . . . . . . . . . . 2017.5.a Derivation of a function discontinuous at a point . . . . . . 2017.5.b Derivative of a function with discontinuity along a surface S 2047.5.c Laplacian of a function discontinuous along a surface S . . 206

7.5.d Application: laplacian of 1/r in 3-space . . . . . . . . . . . . . 2077.6 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2097.6.a The tensor product of two functions . . . . . . . . . . . . . . 2097.6.b The tensor product of distributions . . . . . . . . . . . . . . . 2097.6.c Convolution of two functions . . . . . . . . . . . . . . . . . . 2117.6.d Fuzzy measurement . . . . . . . . . . . . . . . . . . . . . . . 2137.6.e Convolution of distributions . . . . . . . . . . . . . . . . . . . 2147.6.f Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2157.6.g The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . 216

7.7 Physical interpretation of convolution operators . . . . . . . . . . . . . 2177.8 Discrete convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8 Distributions II 2238.1 Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.1.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.1.b Application to the computation of certain integrals . . . . . . 2248.1.c Feynmans notation . . . . . . . . . . . . . . . . . . . . . . . . 2258.1.d Kramers-Kronig relations . . . . . . . . . . . . . . . . . . . . . 2278.1.e A few equations in the sense of distributions . . . . . . . . . . 229

8.2 Topology in D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.2.a Weak convergence in D . . . . . . . . . . . . . . . . . . . . . . 2308.2.b Sequences of functions converging to . . . . . . . . . . . . . 2318.2.c Convergence in D and convergence in the sense of functions 234

8.2.d Regularization of a distribution . . . . . . . . . . . . . . . . . 2348.2.e Continuity of convolution . . . . . . . . . . . . . . . . . . . . 2358.3 Convolution algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2368.4 Solving a differential equation with initial conditions . . . . . . . . . 238

8.4.a First order equations . . . . . . . . . . . . . . . . . . . . . . . . 2388.4.b The case of the harmonic oscillator . . . . . . . . . . . . . . . 2398.4.c Other equations of physical origin . . . . . . . . . . . . . . . . 240



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9 Hilbert spaces; Fourier series 2499.1 Insufficiency of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 2499.2 Pre-Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.2.a The finite-dimensional case . . . . . . . . . . . . . . . . . . . . 2549.2.b Projection on a finite-dimensional subspace . . . . . . . . . . 2549.2.c Bessel inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 256

9.3 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2569.3.a Hilbert basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2579.3.b The 2 space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.3.c The space L2 [0, a] . . . . . . . . . . . . . . . . . . . . . . . . . 2629.3.d The L2() space . . . . . . . . . . . . . . . . . . . . . . . . . . 263

9.4 Fourier series expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2649.4.a Fourier coefficients of a function . . . . . . . . . . . . . . . . 2649.4.b Mean-square convergence . . . . . . . . . . . . . . . . . . . . . 265

9.4.c Fourier series of a function f L1

[0, a] . . . . . . . . . . . . 2669.4.d Pointwise convergence of the Fourier series . . . . . . . . . . . 2679.4.e Uniform convergence of the Fourier series . . . . . . . . . . . 2699.4.f The Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . 270


10 Fourier transform of functions 27710.1 Fourier transform of a function in L1 . . . . . . . . . . . . . . . . . . 277

10.1.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810.1.b Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

10.1.c The L1 space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27910.1.d Elementary properties . . . . . . . . . . . . . . . . . . . . . . . 28010.1.e Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28210.1.f Extension of the inversion formula . . . . . . . . . . . . . . . 284

10.2 Properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . 28510.2.a Transpose and translates . . . . . . . . . . . . . . . . . . . . . . 28510.2.b Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28610.2.c Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28610.2.d Rapidly decaying functions . . . . . . . . . . . . . . . . . . . . 288

10.3 Fourier transform of a function in L2 . . . . . . . . . . . . . . . . . . 28810.3.a The space S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10.3.b The Fourier transform in L

2

. . . . . . . . . . . . . . . . . . . 29010.4 Fourier transform and convolution . . . . . . . . . . . . . . . . . . . . 29210.4.a Convolution formula . . . . . . . . . . . . . . . . . . . . . . . 29210.4.b Cases of the convolution formula . . . . . . . . . . . . . . . . 293


11 Fourier transform of distributions 29911.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . 299

11.1.a Tempered distributions . . . . . . . . . . . . . . . . . . . . . . 30011.1.b Fourier transform of tempered distributions . . . . . . . . . . 30111.1.c Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303


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11.1.d Higher-dimensional Fourier transforms . . . . . . . . . . . . . 30511.1.e Inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.2 The Dirac comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

11.2.a Definition and properties . . . . . . . . . . . . . . . . . . . . . 30711.2.b Fourier transform of a periodic function . . . . . . . . . . . . 30811.2.c Poisson summation formula . . . . . . . . . . . . . . . . . . . 30911.2.d Application to the computation of series . . . . . . . . . . . . 310

11.3 The Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.4 Application to physical optics . . . . . . . . . . . . . . . . . . . . . . . 314

11.4.a Link between diaphragm and diffraction figure . . . . . . . . 31411.4.b Diaphragm made of infinitely many infinitely narrow slits . 31511.4.c Finite number of infinitely narrow slits . . . . . . . . . . . . . 31611.4.d Finitely many slits with finite width . . . . . . . . . . . . . . . 31811.4.e Circular lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

11.5 Limitations of Fourier analysis and wavelets . . . . . . . . . . . . . . . 321Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

12 The Laplace transform 33112.1 Definition and integrability . . . . . . . . . . . . . . . . . . . . . . . . 331

12.1.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.1.b Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.1.c Properties of the Laplace transform . . . . . . . . . . . . . . . 336

12.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33612.3 Elementary properties and examples of Laplace transforms . . . . . . 338

12.3.a Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33812.3.b Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33912.3.c Differentiation and integration . . . . . . . . . . . . . . . . . . 33912.3.d Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

12.4 Laplace transform of distributions . . . . . . . . . . . . . . . . . . . . 34212.4.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34212.4.b Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34212.4.c Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34412.4.d The z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 34412.4.e Relation between Laplace and Fourier transforms . . . . . . . 345

12.5 Physical applications, the Cauchy problem . . . . . . . . . . . . . . . 346

12.5.a Importance of the Cauchy problem . . . . . . . . . . . . . . . 34612.5.b A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . 34712.5.c Dynamics of the electromagnetic field without sources . . . . 348


13 Physical applications of the Fourier transform 35513.1 Justification of sinusoidal regime analysis . . . . . . . . . . . . . . . . 35513.2 Fourier transform of vector fields: longitudinal and transverse fields 35813.3 Heisenberg uncertainty relations . . . . . . . . . . . . . . . . . . . . . 35913.4 Analytic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36513.5 Autocorrelation of a finite energy function . . . . . . . . . . . . . . . 368


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13.5.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36813.5.b Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36813.5.c Intercorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

13.6 Finite power functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37013.6.a Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37013.6.b Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

13.7 Application to optics: the Wiener-Khintchine theorem . . . . . . . . 371Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

14 Bras, kets, and all that sort of thing 37714.1 Reminders about finite dimension . . . . . . . . . . . . . . . . . . . . 377

14.1.a Scalar product and representation theorem . . . . . . . . . . . 37714.1.b Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

14.1.c Symmetric and hermitian endomorphisms . . . . . . . . . . . 37914.2 Kets and bras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37914.2.a Kets | H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37914.2.b Bras | H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38014.2.c Generalized bras . . . . . . . . . . . . . . . . . . . . . . . . . . 38214.2.d Generalized kets . . . . . . . . . . . . . . . . . . . . . . . . . . 38314.2.e Id =

n |n n| . . . . . . . . . . . . . . . . . . . . . . . . . 384

14.2.f Generalized basis . . . . . . . . . . . . . . . . . . . . . . . . . . 38514.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

14.3.a Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38714.3.b Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38914.3.c Bounded operators, closed operators, closable operators . . . 390

14.3.d Discrete and continuous spectra . . . . . . . . . . . . . . . . . 39114.4 Hermitian operators; self-adjoint operators . . . . . . . . . . . . . . . 393

14.4.a Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39414.4.b Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39614.4.c Generalized eigenvectors . . . . . . . . . . . . . . . . . . . . . . 39714.4.d Matrix representation . . . . . . . . . . . . . . . . . . . . . . 39814.4.e Summary of properties of the operators P and X . . . . . . . 401


15 Green functions 40715.1 Generalities about Green functions . . . . . . . . . . . . . . . . . . . . 40715.2 A pedagogical example: the harmonic oscillator . . . . . . . . . . . . 409

15.2.a Using the Laplace transform . . . . . . . . . . . . . . . . . . . 41015.2.b Using the Fourier transform . . . . . . . . . . . . . . . . . . . 410

15.3 Electromagnetism and the dAlembertian operator . . . . . . . . . . . 41415.3.a Computation of the advanced and retarded Green functions 41415.3.b Retarded potentials . . . . . . . . . . . . . . . . . . . . . . . . . 41815.3.c Covariant expression of advanced and retarded Green functions 42115.3.d Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

15.4 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42215.4.a One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 42315.4.b Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . 426


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15.5 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42715.6 Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

16 Tensors 43316.1 Tensors in affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

16.1.a Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43316.1.b Einstein convention . . . . . . . . . . . . . . . . . . . . . . . . 43516.1.c Linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43616.1.d Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43816.1.e Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . 439

16.2 Tensor product of vector spaces: tensors . . . . . . . . . . . . . . . . . 43916.2.a Existence of the tensor product of two vector spaces . . . . . 43916.2.b Tensor product of linear forms: tensors of type

0

2. . . . . . 441

16.2.c Tensor product of vectors: tensors of type 20 . . . . . . . . . 44316.2.d Tensor product of a vector and a linear form: linear maps

or1

1

-tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

16.2.e Tensors of typep

q

. . . . . . . . . . . . . . . . . . . . . . . . . 446

16.3 The metric, or, how to raise and lower indices . . . . . . . . . . . . . 44716.3.a Metric and pseudo-metric . . . . . . . . . . . . . . . . . . . . . 44716.3.b Natural duality by means of the metric . . . . . . . . . . . . . 44916.3.c Gymnastics: raising and lowering indices . . . . . . . . . . . . 450

16.4 Operations on tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 45316.5 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

16.5.a Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . 45516.5.b Basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45616.5.c Transformation of physical quantities . . . . . . . . . . . . . . 45816.5.d Transformation of linear forms . . . . . . . . . . . . . . . . . . 45916.5.e Transformation of an arbitrary tensor field . . . . . . . . . . . 46016.5.f Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

17 Differential forms 46317.1 Exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

17.1.a 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46317.1.b Exterior 2-forms . . . . . . . . . . . . . . . . . . . . . . . . . . 464

17.1.c Exterior k-forms . . . . . . . . . . . . . . . . . . . . . . . . . . 46517.1.d Exterior product . . . . . . . . . . . . . . . . . . . . . . . . . . 467

17.2 Differential forms on a vector space . . . . . . . . . . . . . . . . . . . 46917.2.a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46917.2.b Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . 470

17.3 Integration of differential forms . . . . . . . . . . . . . . . . . . . . . . 47117.4 Poincars theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47417.5 Relations with vector calculus: gradient, divergence, curl . . . . . . . 476

17.5.a Differential forms in dimension 3 . . . . . . . . . . . . . . . . 47617.5.b Existence of the scalar electrostatic potential . . . . . . . . . . 47717.5.c Existence of the vector potential . . . . . . . . . . . . . . . . . 47917.5.d Magnetic monopoles . . . . . . . . . . . . . . . . . . . . . . . . 480


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17.6 Electromagnetism in the language of differential forms . . . . . . . . 480Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

18 Groups and group representations 48918.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48918.2 Linear representations of groups . . . . . . . . . . . . . . . . . . . . . 49118.3 Vectors and the group SO(3) . . . . . . . . . . . . . . . . . . . . . . . 49218.4 The group SU(2) and spinors . . . . . . . . . . . . . . . . . . . . . . . 49718.5 Spin and Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . 503

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

19 Introduction to probability theory 50919.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

19.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51219.3 Poincar formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51619.4 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . 51719.5 Independent events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

20 Random variables 52120.1 Random variables and probability distributions . . . . . . . . . . . . 52120.2 Distribution function and probability density . . . . . . . . . . . . . 524

20.2.a Discrete random variables . . . . . . . . . . . . . . . . . . . . . 52620.2.b (Absolutely) continuous random variables . . . . . . . . . . . 526

20.3 Expectation and variance . . . . . . . . . . . . . . . . . . . . . . . . . 52720.3.a Case of a discrete r.v. . . . . . . . . . . . . . . . . . . . . . . . . 527

20.3.b Case of a continuous r.v. . . . . . . . . . . . . . . . . . . . . . 52820.4 An example: the Poisson distribution . . . . . . . . . . . . . . . . . . 53020.4.a Particles in a confined gas . . . . . . . . . . . . . . . . . . . . . 53020.4.b Radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . 531

20.5 Moments of a random variable . . . . . . . . . . . . . . . . . . . . . . 53220.6 Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

20.6.a Pair of random variables . . . . . . . . . . . . . . . . . . . . . . 53420.6.b Independent random variables . . . . . . . . . . . . . . . . . . 53720.6.c Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 538

20.7 Image measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53920.7.a Case of a single random variable . . . . . . . . . . . . . . . . . 53920.7.b Case of a random vector . . . . . . . . . . . . . . . . . . . . . . 540

20.8 Expectation and characteristic function . . . . . . . . . . . . . . . . . 54020.8.a Expectation of a function of random variables . . . . . . . . 54020.8.b Moments, variance . . . . . . . . . . . . . . . . . . . . . . . . . 54120.8.c Characteristic function . . . . . . . . . . . . . . . . . . . . . . 54120.8.d Generating function . . . . . . . . . . . . . . . . . . . . . . . . 543

20.9 Sum and product of random variables . . . . . . . . . . . . . . . . . . 54320.9.a Sum of random variables . . . . . . . . . . . . . . . . . . . . . 54320.9.b Product of random variables . . . . . . . . . . . . . . . . . . . 54620.9.c Example: Poisson distribution . . . . . . . . . . . . . . . . . . 547

20.10 Bienaym-Tchebychev inequality . . . . . . . . . . . . . . . . . . . . . 54720.10.a Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547


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20.10.b Application: Buffons needle . . . . . . . . . . . . . . . . . . . 54920.11 Independance, correlation, causality . . . . . . . . . . . . . . . . . . . 550

21 Convergence of random variables: central limit theorem 55321.1 Various types of convergence . . . . . . . . . . . . . . . . . . . . . . . . 55321.2 The law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . 55521.3 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

Appendices

A Reminders concerning topology and normed vector spaces 573A.1 Topology, topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 573A.2 Normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

A.2.a Norms, seminorms . . . . . . . . . . . . . . . . . . . . . . . . . 577A.2.b Balls and topology associated to the distance . . . . . . . . . . 578A.2.c Comparison of sequences . . . . . . . . . . . . . . . . . . . . . 580A.2.d Bolzano-Weierstrass theorems . . . . . . . . . . . . . . . . . . . 581A.2.e Comparison of norms . . . . . . . . . . . . . . . . . . . . . . . 581A.2.f Norm of a linear map . . . . . . . . . . . . . . . . . . . . . . . 583

Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584B Elementary reminders of differential calculus 585

B.1 Differential of a real-valued function . . . . . . . . . . . . . . . . . . . 585B.1.a Functions of one real variable . . . . . . . . . . . . . . . . . . 585B.1.b Differential of a function f : n . . . . . . . . . . . . . 586B.1.c Tensor notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

B.2 Differential of map with values in p . . . . . . . . . . . . . . . . . . 587B.3 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

C Matrices 593

C.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593C.2 Application to matrix representation . . . . . . . . . . . . . . . . . . . 594

C.2.a Matrix representing a family of vectors . . . . . . . . . . . . . 594C.2.b Matrix of a linear map . . . . . . . . . . . . . . . . . . . . . . 594C.2.c Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 595C.2.d Change of basis formula . . . . . . . . . . . . . . . . . . . . . 595C.2.e Case of an orthonormal basis . . . . . . . . . . . . . . . . . . 596

D A few proofs 597


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Contents x v

Tables

Fourier transforms 609

Laplace transforms 613

Probability laws 616

Further reading 617

References 621

Portraits 627

Sidebars 629

Index 631


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17/664

To Kirone Mallick, Angel Alastuey and Michael Kiessling,for their friendship and our discussions scientific or otherwise.

Thanks

Many are those who I wish to thank for this book: Jean-Franois Colombeauwho was in charge of the course of Mathematics for the Magistre des sciences de lamatire of the ENS Lyon before me; Michel Peyrard who then asked me to replacehim; Vronique Terras and Jean Farago who were in charge of the exercise sessionsduring the three years when I taught this course.

Many friends and colleagues were kind enough to read and re-read one or more

chapters: Julien Barr, Maxime Clusel, Thierry Dauxois, Kirone Mallick, JulienMichel, Antoine Naert, Catherine Ppin, Magali Ribaut, Erwan Saint-Loubert Bi;and especially Paul Pichaureau, TEX-guru, who gave many (very) sharp and (always)pertinent comments concerning typography and presentation as well as contents. Ialso wish to thank the ditions H&K and in particular Sbastien Desreux, for hishard work and stimulating demands; Brangre Condomines for her attentive reading.

I am also indebted to Jean-Franois Quint for many long and fascinating mathe-matical discussions.

This edition owes a lot to Emmanuel Kowalski, friend of many years, Professor atthe University of Bordeaux, who led me to refine certain statements, correct others,and more generally helped to clarify some delicate points.

For sundry diverse reasons, no less important, I want to thank Craig Thompson,

author ofGood-bye Chunky Rice[89]and Blankets [90], who created the illustrations forChapter 1; Jean-Paul Marchal, master typographer and maker of popular images inpinal; Angel Alastuey who taught me so much in physics and mathematics; Frdric,Latitia, Samuel, Koupaa and Alanis Vivien for their support and friendship; ClaudeGarcia, for his joie de vivre, his fideos, his fine bottles, his knowledge of topology, hisadvice on Life, the Universe... and the Rest! without forgetting Anita, Alice and Hugofor their affection.

I must not forget the many readers who, since the first French edition, havecommunicated their remarks and corrections; in particular Jean-Julien Fleck, MarcRezzouk, Franoise Cornu, Cline Chevalier and professors Jean Cousteix, Andreasde Vries, and Franois Thirioux; without forgetting all the students of the Magistre

des sciences de la matire of ENS Lyon, promotions 1993 to 1999, who followedmy classes and exercise sessions and who contributed, through their remarks andquestions, to the elaboration and maturation of this book.

Et, bien sr, les plus importants, Anne-Julia, Solveig et Anton avec tout monamour.

Although the text has been very consciously written, read, and proofread, errors, omissions or impreci-sions may still remain. The author welcomes any remark, criticism, or correction that a reader may wish tocommunicate, care of the publisher, for instance, by means of the email address [email protected].

A list of errata will be kept updated on the web site

http://press.princeton.edu


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A books apology

Why should a physicist study mathematics? There is a fairly fashion-able current of thought that holds that the use of advanced mathematics is oflittle real use in physics, and goes sometimes as far as to say that knowingmathematics is already by itself harmful to true physics. However, I am and remainconvinced that mathematics is still a precious source of insight, not only for students ofphysics, but also for researchers.

Many only see mathematics as a tool and of course, it is in part a tool, butthey should be reminded that, as Galileo said, the book of Nature is written in themathematicians language.1 Since Galileo and Newton, the greatest physicists giveexamples that knowing mathematics provides the means to understand precise physicalnotions, to use them more easily, to establish them on a sure foundation, and evenmore importantly, to discover new ones.2 In addition to ensuring a certain rigor inreasoning which is indispensable in any scientific study, mathematics belongs to the

natural lexicon of physicists. Even if the rules of proportion and the fundamentals ofcalculus are sufficient for most purposes, it is clear that a richer vocabulary can leadto much deeper insights. Imagine if Shakespeare or Joyce had only had a few hundredwords to chose from!

Is is therefore discouraging to sometimes hear physicists dismiss certain theories be-cause this is only mathematics. In fact, the two disciplines are so closely related thatthe most prestigious mathematical award, the Fields medal, was given to the physicistEdwardWitten in 1990, rewarding him for the remarkable mathematical discoveriesthat his ideas led to.

How should you read this book? Or rather how should you not read some parts?

1 Philosophy is written is this vast book which is forever open before our eyes I mean, the Universe but it can not be read until you have learned the tongue and understood the character in which it iswritten. It is written in the language of mathematics, and its characters are triangles, circles, and other

geometric pictures, without which it is not humanly possible to understand a single word[...].2 I will only mention Newton (gravitation, differential and integral calculus), Gauss (optics,

magnetism, all the mathematics of his time, and quite a bit that was only understood muchlater), Hamilton (mechanics, differential equations, algebra), Heaviside (symbolic calculus,signal theory), Gibbs (thermodynamics, vector analysis) and of course Einstein. One couldwrite a much longer list. If Richard Feynman presents a very physical description of his artin his excellent physics course [35], appealing to remarkably little formalism, it is neverthelessthe fact that he was the master of the elaborate mathematics involved, as his research works

show.


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A books apology x i x

Since the reader may want to learn first certain specific topics among those present, hereis a short description of the contents of this text:

The first chapter serves as a reminder of some simple facts concerning the funda-mental notion of convergence. There shouldnt be much in the way of newmathematics there for anyone who has followed a rigorous calculus course,but there are many counter-examples to be aware of. Most importantly, a longsection describes the traps and difficulties inherent in the process of exchangingtwo limits in the setting of physical models. It is not always obvious where,in physical reasoning, one has to exchange two mathematical limits, and manyapparent paradoxes follow from this fact.

The real beginning concerns the theory of integration, which is briefly presentedin the form of the Lebesgue integral based on measure theory (Chapter 2). For

many readers, this may be omitted in a first reading. Chapter 3 discusses thebasic results and techniques of integral calculus.

Chapters 4 to 6 present the theory of functions of a complex variable, with anumber of applications:

the residue method, which is an amazing tool for integral calculus;

some physical notions, such as causality, are very closely related to analyt-icity of functions on the complex plane (Section 13.4);

harmonic functions (such thatf = 0) in two dimensions are linked tothe real parts of holomorphic (analytic) functions (Chapter 5);

conformal maps (those which preserve angles) can be used to simplifyboundary conditions in problems of hydrodynamics or electromagnetism(Chapter 6);

Chapters 7 and 8 concern the theory of distributions (generalized functions)and their applications to physics. These form a relatively self-contained subset ofthe book.

Chapters 9 to 12 deal with Hilbert spaces, Fourier series, Fourier and Laplacetransforms, which have too many physical applications to attempt a list. Chap-ter 13 develops some of those applications, and this chapter also requires complex

analysis. Chapter 14 is a short (probably too short) introduction to the Dirac notation

used in quantum mechanics: kets | andbras |. The notions of generalizedeigenbasis and self-adjoint operators on Hilbert space are also discussed.

Several precise physical problems are considered and solved in Chapter 15 by themethod of Green functions. This method is usually omitted from textbooks onelectromagnetism (where a solution is taken straight out of a magicians hat) orof field theory (where it is assumed that the method is known). I hope to filla gap for students by presenting the necessary (and fairly simple) computations

from beginning to end, using physicists notation.


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x x A books apology

Chapters 16 and 17 about tensor calculus and differential forms are also some-what independent from the rest of the text. Those two chapters are only briefintroductions to their respective topics.

Chapter 18 has the modest goal of relating some notions of topology and grouptheory to the idea of spin in quantum mechanics.

Probability theory, discussed in Chapters 19 to 21, is almost entirely absentfrom the standard physics curriculum, although the basic vocabulary and re-sults of probability seem necessary to any physicist interested in theory (stochasticequations, Brownian motion, quantum mechanics and statistical mechanics allrequire probability theory) or experiments (Gaussian white noise, measurementerrors, standard deviation of a data set...)

Finally, a few appendices contain further reminders of elementary mathematicalnotions and the proofs of some interesting results, the length of which made theirinsertion in the main text problematic.

Many physical applications, using mathematical tools with the usual nota-tion of physics, are included in the text. They can be found by looking in theindex at the items under Physical applications.


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A books apology x x i

Remarks on the biographical summaries

The short biographies of mathematicians which are interspersed in the text are taken from

many sources: Bertrand Hauchecorne and Daniel Suratteau, Des mathmaticiens de A Z, Ellipses,1996 (French).

The web site of Saint-Andrews University (Scotland)www-history.mcs.st-andrews.ac.uk/history/Mathematicians

and the web site of the University of Colorado at Boulder

www.colorado.edu/education/DMP

from which certain pictures are also taken.

The literary structure of scientific argument, edited by Peter Dear, University of PennsylvaniaPress, 1991. Simon Gindikin, Histoires de mathmaticiens et de physiciens, Cassini, 2000. Encyclopdia Universalis, Paris, 1990.

Translators foreword

I am a mathematician and have now forgotten most of the little physics I learned in school(although Ive probably picked up a little bit again by translating this book). I would like tomention here two more reasons to learn mathematics, and why this type of book is thereforevery important.

First, physicists benefit from knowing mathematics (in addition to the reasons Waltermentioned) because, provided they immerse themselves in mathematics sufficiently to becomefluent in its language, they will gain access to newintuitions. Intuitions are very different fromany set of techniques, or tools, or methods, but they are just as indispensable for a researcher,and they are the hardest to come by.3 A mathematicians intuitions are very different fromthose of a physicist, and to have both available is an enormous advantage.

The second argument is different, and may be subjective: physics is hard, much harderin some sense than mathematics. A very simple and fundamental physical problem maybe all but impossible to solve because of the complexity (apparent or real) of Nature. Butmathematicians know that a simple, well-formulated, natural mathematical problem (in somesense that is impossible to quantify!) has a simple solution. This solution may require inventingentirely new concepts, and may have to wait for a few hundred years before the idea comes toa brilliant mind, but it is there. What this means is that if you manage to put the physicalproblem in a very natural mathematical form, the guiding principles of mathematics may leadyou to the solution. Dirac was certainly a physicist with a strong sense of such possibilities;this led him to discover antiparticles, for instance.

3 Often, nothing will let you understand the intuition behind some important idea except,

essentially, rediscovering by yourself the most crucial part of it.


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Notation

Symbol Meaning Page

\ without (example: A \B) has for Laplace transform 333

, duality bracket: a, x = a(x)(|) scalar product 251

|

,

|ket, bra 379,380

integral partX Dirac comb 1861 constant function x 1 convolution product 211 tensor product 210, 439 exterior product 464 boundary (example: , boundary of) laplacian dAlembertian 414 equivalence of sequences or functions 580

isomorphic to (example: E

F)

approximate equality (in physics)def= equality defining the left-hand side

[a, b] closed interval : {x ; a x b}]a, b[ open interval : {x ; a < x < b}[[1, n]] {1 , . . . , n}


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Index of notation x x i i i

Latin letter Meaning Page

B Borel -algebra 58

B(a ; r), B(a ; r) open (closed) ball centered at a with radius rBil(EF, G) space of bilinear maps from EF to GA complement of A in 512

{} 146C(I, ) vector space of continuous real-valued functions on IC(a ; r) circle centered at a with radius r

Cnt any constantD vector space of test functions 182D vector space of distributions 183

D+, D vector space of distributions with bounded support 236on the left (right)

diag(1, . . . ,

n) square matrix with coefficients m

i j=

i

i je Nepers constant (e: electric charge)(e) basis of a vector space 433E(X) expectation of the r.v. X 527

F[ f], f Fourier transform of f 278f Laplace transform of f 333f transpose of f; f(x) = f(x) 189

f(n) n-th derivative of fGLn() group of invertible matrices of order n over G(E) group of automorphisms of the vector space E

H Heaviside distribution or function 193

, or 2 space of sequences (un)n such that |un|2 < + 261L1 space of integrable functions 280L2 space of square-integrable functions 262

L(E, F) space of linear maps from E to FL Lebesgue -algebra 60

Mn() algebra of square matrices of order n over KP(A) set of subsets of A 57P(A) probability of the event A 514

{, +}Re(z), Im(z) real or imaginary part of z

S space of Schwartz functions 289S space of tempered distributions 300S, Sn group of permutations (ofn elements) 26

t transpose of a matrix : (tA)i j = Aj iu, v, w, x vectors in E 434

pv principal value 188X, Y, Z random variables 521


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x x i v Index of notation

Greek letter

() dual basis of the basis (e) 436

Euler function 154, Dirac distribution, its derivative 185, 196

, i j Kronecker symbol 445() signature of the permutation 465k space of exterior k-forms 464 rectangle function 213, 279

(X) standard deviation of the r.v. X 528

Acharacteristic function of A 63

, differential forms 436

Abbreviations

(n).v.s. (normed) vector spacer.v. random variable 521

i.r.v. independent random variables 537


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Chapter

1Reminders: convergence of

sequences and series

This first chapter, which is quite elementary, is essentially a survey of the notionof convergence of sequences and series. Readers who are very confortable with thisconcept may start reading the next chapter.

However, although the mathematical objects we discuss are well known in princi-ple, they have some unexpected properties. We will see in particular that the orderof summation may be crucial to the evaluation of the series, so that changing theorder of summation may well change its sum.

We start this chapter by discussing two physical problems in which a limit processis hidden. Each leads to an apparent paradox, which can only be resolved when theunderlying limit is explicitly brought to light.

1.1

The problem of limits in physics

1.1.a Two paradoxes involving kinetic energy

First paradox

Consider a truck with mass m driving at constant speed v = 60 mph on aperfectly straight stretch of highway (think of Montana). We assume that,

friction being negligible, the truck uses no gas to remain at constant speed.


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2 Reminders concerning convergence

On the other hand, to accelerate, it must use (or guzzle) gallons of gas inorder to increase its kinetic energy by an amount of 1 joule. This assumption,although it is imperfect, is physically acceptable because each gallon of gas

yields the same amount of energy.So, when the driver decides to increase its speed to reach v = 80 mph, the

quantity of gas required to do so is equal to the difference of kinetic energy,namely, it is

(Ec Ec) =1

2m(v2 v2) = 1

2m(6400 3600) = 1400 m.

With m = 110000

J mile2 hour2, say, this amounts to 0.14 gallon. Jollygood.

Now, let us watch the same scene of the truck accelerating, as observed bya highway patrolman, initially driving as fast as the truck w = v = 60 mph,but with a motorcycle which is unable to go faster.

The patrolman, having his college physics classes at his fingertips, argues asfollows: in my own Galilean reference frame, the relative speed of the truck

was previouslyv = 0 and is now v = 20 mph. To do this, the amount ofgas it has guzzled is equal to the difference in kinetic energies:

(Ec Ec ) =1

2m

(v)2 (v)2

=1

2m(400 0) = 200 m,

or around 0.02 gallons.

There is here a clear problem, and one of the two observers must be wrong.Indeed, the Galilean relativity principle states that all Galilean reference framesare equivalent, and computing kinetic energy in the patrolmans referenceframe is perfectly legitimate.

How is this paradox resolved?We will come to the solution, but first here is another problem. The reader,

before going on to read the solutions, is earnestly invited to think and try tosolve the problem by herself.

Second paradox

Consider a highly elastic rubber ball in free fall as we first see it. At somepoint, it hits the ground, and we assume that this is an elastic shock.

Most high-school level books will describe the following argument: as-sume that, at the instant t = 0 when the ball hits the ground, the speed ofthe ball is v1 = 10 ms1. Since the shock is elastic, there is conservation oftotal energy before and after. Hence the speed of the ball after the rebound isv2 = v1, or simply+10 ms1 going up.

This looks convincing enough. But it is not so impressive if seen fromthe point of view of an observer who is also moving down at constant speedvobs = v1 = 10 ms1. For this observer, the speed of the ball before theshock is v1 = v1 vobs = 0 ms

1

, so it has zero kinetic energy. However,


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The problem of limits in physics 3

after rebounding, the speed of the ball is v2 = v2 vobs = 20 ms1, andtherefore it has nonzero kinetic energy! With the analogue of the reasoningabove, one should still have found v

2= v

1= 0 (should the ball go through

the ground?)So there is something fishy in this argument also. It is important to

remember that the fact that the right answer is found in the first case does not implythat the argument that leads to the answer is itself correct.

Readers who have solved the first paradox will find no difficulty in thissecond one.

Paradoxes resolved

Kinetic energy is of course not the same in every reference frame. But this

is not so much the kinetic energy we are interested in; rather, we want thedifference before and after the event described.Lets go back to elementary mechanics. What happens, in two distinct

reference frames, to a system of N solid bodies with initial speed vi (i =1 , . . . , N) and final speed vi after some shock?

In the first reference frame, the difference of kinetic energy is given by

Ec =Ni=1

mi(vi2 v2i).

In a second reference frame, with relative speed w with respect to the first,

the difference is equal to

Ec =Ni=1

mi

(vi w)2 (v i w)2

=Ni=1

mi(vi2 vi2) 2w

N

i=1

mi(vi vi)

= Ec 2w P ,

(we use as labels for any physical quantity expressed in the new referenceframe), so that Ec = Ec as long as the total momentum is preserved during

the shock, in other words ifP = 0.In the case of the truck and the patrolman, we did not really take themomentum into account. In fact, the truck can accelerate because it pushesback the whole earth behind it!

So, let us take up the computation with a terrestrial mass M, which islarge but not infinite. We will take the limit [M ] at the very end of thecomputation, and more precisely, we will let [M/m ].

At the beginning of the experiment, in the terrestrial reference frame,the speed of the truck is v. At the end of the experiment, the speed is v.Earth, on the other hand, has original speed V = 0, and if one remainsin the same Galilean reference frame, final speed V = m

M

(v

v) (because


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of conservation of total momentum).1 The kinetic energy of the system atthe beginning is then 1

2mv2 and at the end it is 1

2mv2 + 1

2MV2. So, the

difference is given byEc =

1

2m(v2 v2) + 1

2

m2

M(v v)2 = 1

2m(v2 v2)

1 + O

mM

.

This is the amount of gas involved! So we see that, up to negligible terms, thefirst argument gives the right answer, namely, 0.14 gallons.

We now come back to the patrolmans frame, moving with speed w withrespect to the terrestrial frame. The initial speed of the truck is v = v w,and the final speed is v = v w. The Earth has initial speed V = wand final speed V = w + m

M(v v). The difference is now:

Ec

=1

2mv2 v2+

1

2MV2 V2

=1

2m(v w)2 1

2m(v w)2 + 1

2M m

M(v v) w

2 12Mw2

=1

2mv2 1

2mv2 + m(v v) w + 1

2

m2

M(v v)2 m(v v) w,

Ec = Ec.

Hence the difference of kinetic energy is preserved, as we expected. So evenin this other reference frame, a correct computation shows that the quantityof gas involved is the same as before.

The patrolmans mistake was to forget the positive term m(v v) w,corresponding to the difference of kinetic energy of the Earth in its Galileanframe. This term does not tend to 0 as [M/m ] !

From the point of view of the patrolmans frame, 0.02 gallons are needed toaccelerate the truck, and the remaining 0.12 gallons are needed to accelerate the Earth!

We can summarize this as a table, where T is the truck and E is the Earth.

Initialspeed

Finalspeed

Ec init. Ec final Ec

T v v 12

mv2 12mv2 m

2 (v2

v2

)

+m2

2M(v v)2E 0 m

M(v v) 0 1

2m2

M(v v)2

T vw v w 12

m(vw)2 12

m(v w)2 m2

(v2 v2)+

m2

2M(v v)2

E w mM

(v v) w 12

Mw2 M2

m

M(v v) w2

1 Note that the terrestrial reference frame is then not Galilean, since the Earth started

moving under the trucks impulsion.


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(Illustration c Craig Thompson 2001)

Fig. 1.1 Romeo, Juliet, and the boat on a lake.

The second paradox is resolved in the same manner: the Earths reboundenergy must be taken into account after the shock with the ball.

The interested reader will find another paradox, relating to optics, in Exer-cise 1.3 on page 43.

1.1.b Romeo, Juliet, and viscous fluids

Here is an example in mechanics where a function f(x) is defined on [0, +[,but limx0+ f(x) = f(0).Let us think of a summer afternoon, which Romeo and Juliet have dedi-

cated to a pleasant boat outing on a beautiful lake. They are sitting on eachside of their small boat, immobile over the waters. Since the atmosphere isconducive to charming murmurs, Romeo decides to go sit by Juliet.

Denote by M the mass of the boat and Juliet together, m that of Romeo,and L the length of the walk from one side of the boat to the other (seeFigures 1.1 and 1.2).

Two cases may be considered: one where the friction of the boat on thelake is negligible (a perfect fluid), and one where it is given by the formulaf = v, where f is the force exerted by the lake on the boat, v the speed ofthe boat on the water, and a viscosity coefficient. We consider the problemonly on the horizontal axis, so it is one-dimensional.

We want to compute how far the boat moves

1. in the case = 0;

2. in the case = 0.Let be this distance.

The first case is very easy. Since no force is exerted in the horizontal plane

on the system boat + Romeo + Juliet, the center of gravity of this system


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(Illustration c Craig Thompson 2001)

Fig. 1.2 Romeo moved closer.

does not move during the experiment. Since Romeo travels the distance Lrelative to the boat, it is easy to deduce that the boat must cover, in theopposite direction, the distance

= mm + M

L .

In the second case, let x(t) denote the position of the boat and y(t) thatof Romeo, relative to the Earth, not to the boat. The equation of movementfor the center of gravity of the system is

Mx+ my = x.We now integrate on both sides between t = 0 (before Romeo starts moving)and t = +. Because of the friction, we know that as [t +], the speedof the boat goes to 0 (hence also the speed of Romeo, since he will have beenlong immobile with respect to the boat). Hence we have

(Mx+ my)+

0= 0 = x(+) x(0)

or = 0. Since = 0, we have = 0, whichever way Romeo moved to theother side. In particular, if we take the limit when 0, hoping to obtainthe nonviscous case, we have:

lim0>0

() = 0 hence lim0>0

() =(0).


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x

V(x)

V0

Fig. 1.3 Potential wall V(x) = V0 H(x).

Conclusion: The limit of a viscous fluid to a perfect fluid is singular. It is not

possible to formally take the limit when the viscosity tends to zero to obtainthe situation for a perfect fluid. In particular, it is easier to model flows ofnonviscous perfect fluids by real fluids which have large viscosity, because ofturbulence phenomena which are more likely to intervene in fluids with small

viscosity. The interested reader can look, for instance, at the book by Guyon,Hulin, and Petit [44].

Remark 1.1 The exact form f = v of the friction term is crucial in this argument. If theforce involves additional (nonlinear) terms, the result is completely different. Hence, if you tryto perform this experiment in practice, it will probably not be conclusive, and the boat is notlikely to come back to the same exact spot at the end.

1.1.c Potential wall in quantum mechanics

In this next physical example, there will again be a situation where we have alimit lim

x0 f(x) = f(0); however, the singularity arises here in fact because ofa second variable, and the true problem is that we have a double limit whichdoes not commute: lim

x0 limy0 f(x, y) = limy0 limx0 f(x, y).The problem considered is that of a quantum particle arriving at a poten-

tial wall. We look at a one-dimensional setting, with a potential of the typeV(x) = V0 H(x), where H is the Heaviside function, that is, H(x) = 0 if

x < 0 and H(x) = 1 if x > 0. The graph of this potential is represented inFigure 1.3.A particle arrives from x = in an energy state E > V0; part of it is

transmitted beyond the potential wall, and part of it is reflected back. We areinterested in the reflection coefficient of this wave.

The incoming wave may be expressed, for negative values of x, as the sumof a progressive wave moving in the direction of increasing x and a reflected

wave. For positive values of x, we have a transmitted wave in the directionof increasing x, but no component in the other direction. According to theSchrdinger equation, the wave function can therefore be written in the form

(x, t) = (x) f(t),


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x

V(x)

V0

a

Fig. 1.4 Smoothed potential V(x) = V0/(1 + ex/a ), with a > 0.

where

(x) =e ik x +B eik x if x < 0, with k

def= 2mEh

,

A e ikx if x > 0, with k def=

2m(EV0)

h.

The function f(t) is only a time-related phase factor and plays no role inwhat follows. The reflection coefficient of the wave is given by the ratio ofthe currents associated with and is given by R = 1 k

k|A|2 (see [20,58]).

Now what is the value of A? To find it, it suffices to write the equationexpressing the continuity of and at x = 0. Since (0+) = (0), wehave 1 +B = A . And since (0+) = (0), we have k(1

B) = kA , and

we deduce that A = 2k/(k + k). The reflection coefficient is therefore equalto

R = 1 k

k|A|2 =

k kk + k

=

EE V0E+

E V0

2. (1.1)

Here comes the surprise: this expression (1.1) is independent of h. In particu-lar, the limit as [h 0] (which defines the classical limit) yields a nonzeroreflection coefficient, although we know that in classical mechanics a particle

with energy E does not reflect against a potential wall with value V0 < E!2

So, displaying explicitly the dependency of R on h, we have:

limh0h=0

R(h) = 0 =R(0).

In fact, we have gone a bit too fast. We take into account the physicalaspects of this story: the classical limit is certainly not the same as brutally

writing h 0. Since Plancks constant is, as the name indicates, just aconstant, this makes no sense. To take the limit h 0 means that onearranges for the quantum dimensions of the system to be much smaller thanall other dimensions. Here the quantum dimension is determined by the de

2

The particle goes through the obstacle with probability 1.


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Broglie wavelength of the particle, that is, = h/p . What are the otherlengths in this problem? Well, there are none! At least, the way we phrased it:because in fact, expressing the potential by means of the Heaviside function israther cavalier. In reality, the potential must be continuous. We can replace itby an infinitely differentiable potential such as V(x) = V0/(1 + e

x/a), whichincreases, roughly speaking, on an interval of size a > 0 (see Figure 1.4). Inthe limit where a 0, the discontinuous Heaviside potential reappears.

Computing the reflection coefficient with this potential is done similarly,but of course the computations are more involved. We refer the reader to [58,chapter 25]. At the end of the day, the reflection coefficient is found todepend not only on h, but also on a, and is given by

R(h, a) = sinh a(k k)sinh a(k + k)

2

.

(h appears in the definition of k =

2mE/h and k =

2m(E V0)/h.)We then see clearly that for fixed nonzero a, the de Broglie wavelength of theparticle may become infinitely small compared to a, and this defines thecorrect classical limit. Mathematically, we have

a = 0 limh0h=0

R(h, a) = 0 classical limit

On the other hand, if we keep h fixed and let a to to 0, we are convergingto the Heaviside potential and we find that

h = 0 lima0a=0

R(h, a) =

k kk + k

2=R(h, 0).

So the two limits [h 0] and [a 0] cannot be taken in an arbitrary order:

limh0h=0

lima0a=0

R(h, a) =

k kk + k

2but lim

a0a=0

limh0h=0

R(h, a) = 0.

To speak ofR(0, 0

)has a priorino physical sense.

1.1.d Semi-infinite filter behaving as waveguide

We consider the circuit A B ,

C C C

A

B

2C

L L L

made up of a cascading sequence of T cells (2C, L, 2C),


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A

B

2C

L

2C

(the capacitors of two successive cells in series are equivalent with one capacitorwith capacitance C). We want to know the total impedance of this circuit.

First, consider instead a circuit made with a finite sequence of n elementarycells, and let Zn denote its impedance. Kirchhoffs laws imply the followingrecurrence relation between the values of Zn :

Zn+1 =

1

2iC +

iL

1

2iC+ Zn

iL + 12iC

+ Zn, (1.2)

where is the angular frequency. In particular, note that if Zn is purelyimaginary, then so is Zn+1. Since Z1 is purely imaginary, it follows that

Zn i for all n .We dont know if the sequence (Zn)n converges. But one thing is cer-

tain: if this sequence (Zn)n converges to some limit, this must be purelyimaginary (the only possible real limit is zero).

Now, we compute the impedance of the infinite circuit, noting that thiscircuit A B is strictly equivalent to the following:

A

B

2C

L Z

2C

Hence we obtain an equation involving Z:

Z =1

2iC+

iL 12iC +ZiL +

1

2iC+ Z

. (1.3)

Some computations yield a second-degree equation, with solutions given by

Z2 =1

C

L 14C2

.

We must therefore distinguish two cases:


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If < c =1

2

L C, we have Z2 < 0 and hence Z is purely imaginary of

the formZ = i 1

4C22 L

C.

Remark 1.2 Mathematically, there is nothing more that can be said, and in particular thereremains an uncertainty concerning the sign of Z.

However, this can also be determined by a physical argument: let tend to 0 (continuousregime). Then we have

Z() 0+

i2C

,

the modulus of which tends to infinity. This was to be expected: the equivalent circuit isopen, and the first capacitor cuts the circuit. Physically, it is then natural to expect that, the

first coil acting as a plain wire, the first capacitor will be dominant. Then

Z() 0+

i2C

(corresponding to the behavior of a single capacitor).Thus the physically acceptable solution of the equation (1.3) is

Z = i

1

4C22 L

C.

If > c =1

2

L C, then Z2 > 0 and Z is therefore real:

Z = L

C 1

4C22.

Remark 1.3 Here also the sign of Z can be determined by physical arguments. The real partof an impedance (the resistive part) is always non-negativein the case of a passive component,since it accounts for the dissipation of energy by the Joule effect. Only active components(such as operational amplifiers) can have negative resistance. Thus, the physically acceptablesolution of equation (1.3) is

Z = +

L

C 1

4C22.

In this last case, there seems to be a paradox since Z cannot be the limitas n + of(Zn)n. Lets look at this more closely.

From the mathematical point of view, Equation (1.3) expresses nothing but thefact that Z is a fixed point for the induction relation (1.2). In other words,this is the equation we would have obtained from (1.2), by continuity, if wehad known that the sequence (Zn)n converges to a limit Z. However, thereis no reason for the sequence (Zn)n to converge.

Remark 1.4 From the physical point of view, the behavior of this infinite chain is rather sur-prising. How does resistive behavior arise from purely inductive or capacitative components?Where does energy dissipate? And where does it go?


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In fact, there is no dissipation of energy in the sense of the Joule effect, but energy doesdisappear from the point of view of an operator holding points A and B. More precisely,one can show that there is a flow of energy propagating from cell to cell. So at the beginning

of the circuit, it looks like there is an energy sink with fixed power consumption. Still, noenergy disappears: an infinite chain can consume energy without accumulating it anywhere.3

In the regime considered, this infinite chain corresponds to a waveguide.

We conclude this first section with a list of other physical situations wherethe problem of noncommuting limits arises:

taking the classical (nonquantum) limit, as we have seen, is by nomeans a trivial matter; in addition, it may be in conflict with a non-relativistic limit (see, e.g., [6]), or with a low temperature limit;

in plasma thermodynamics, the limit of infinite volume ( V

) and

the nonrelativistic limit (c ) are incompatible with the thermo-dynamic limit, since a characteristic time of return to equilibrium isV1/3/c;

in the classical theory of the electron, it is often said that such a classicalelectron, with zero naked mass, rotates too fast at the equator (200 timesthe speed of light) for its magnetic moment and renormalized mass toconform to experimental data. A more careful calculation by Lorentzhimself4 gave about 10 times the speed of light at the equator. But infact, the limit [m 0+] requires care, and if done correctly, it imposesa limit [v/c

1] to maintain a constant renormalized mass [7];

another interesting example is an infinite universe limit and a diluteduniverse limit [56].

1.2

Sequences

1.2.a Sequences in a normed vector space

We consider in this section a normed vector space (E, ) and sequences ofelements of E.5

DEFINITION 1.5 (Convergence of a sequence) Let (E, ) be a normed vec-tor space and (un)n a sequence of elements of E, and let E. The

3 This is the principle of Hilberts infinite hotel.4 Pointed out by Sin-Itiro Tomonaga [91].5

We recall the basic definitions concerning normed vector spaces in Appendix A.


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Sequences 13

sequence (un)n converges to if, for any > 0, there exists an indexstarting from which un is at most at distance from :

> 0 N n n N = un < .Then is called the limit of the sequence (un )n, and this is denoted

= limn un or un n .

DEFINITION 1.6 A sequence (un)n of real numbers converges to + (resp.to ) if, for any M , there exists an index N, starting from which allelements of the sequence are larger than M:

M N n n N = un > M (resp. un < M).

In the case of a complex-valued sequence, a type of convergence to infinity,in modulus, still exists:

DEFINITION 1.7 A sequence (zn)n of complex numbers converges to in-finity if, for any M , there exists an index N, starting from which allelements of the sequence have modulus larger than M:

M N n n N = |zn| > M.Remark 1.8 There is only one direction to infinity in . We will see a geometric interpreta-tion of this fact in Section 5.4 on page 146.

Remark 1.9 The strict inequalities un < (or |un| > M) in the definitions above (which,in more abstract language, amount to an emphasis on open subsets) may be replaced byun , which are sometimes easier to handle. Because > 0 is arbitrary, this givesan equivalent definition of convergence.

1.2.b Cauchy sequences

It is often important to show that a sequence converges, without explicitlyknowing the limit. Since the definition of convergence depends on the limit ,it is not conveninent for this purpose.6 In this case, the most common toolis the Cauchy criterion, which depends on the convergence of elements of the

sequence with respect to each other:

DEFINITION 1.10 (Cauchy criterion) A sequence (un)n in a normed vectorspace is a Cauchy sequence, or satisfies the Cauchy criterion, if

> 0 N p , q q> p N = up uq < 6 As an example: how should one prove that the sequence (un)n, with

un =n

p=1

(1)p+1p4

,

converges? Probably not by guessing that the limit is 74/720.


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or, equivalently, if

> 0

N

p , k

p N =

up+k up < .A common technique used to prove that a sequence (un)n is a Cauchysequence is therefore to find a sequence (p )p of real numbers such that

limp p = 0 and p , k

up+k up p .PROPOSITION 1.11 Any convergent sequence is a Cauchy sequence.

This is a trivial consequence of the definitions. But we are of courseinterested in the converse. Starting from the Cauchy criterion, we want to be

able to conclude that a sequence converges without, in particular, requiringthe limit to be known beforehand. However, that is not always possible:there exist normed vector spaces E and Cauchy sequences in E which do notconverge.

Example 1.12 Consider the set of rational numbers . With the absolute value, it is a normed-vector space. Consider then the sequence

u0 = 3 u1 = 3.1 u2 = 3.14 u3 = 3.141 u4 = 3.1415 u5 = 3.14159 (you can guess the rest7...). This is a sequence of rationals, which is a Cauchy sequence (thedistance between up and up+k is at most 10

p ). However, it does not converge in , since itslimit (in !) is , which is a notoriously irrational number.

The space is not nice in the sense that it leaves a lot of room for Cauchy sequences toexist without converging in . The mathematical terminology is that is not complete.

DEFINITION 1.13 (Complete vector space) A normed vector space (E, ) iscomplete if all Cauchy sequences in E are convergent.

THEOREM 1.14 The spaces and, and more generally all finite-dimensional realor complete normed vector spaces, are complete.

Proof.First case: It is first very simple to show that a Cauchy sequence (un)n of real num-

bers is bounded. Hence, according to the Bolzano-Weierstrass theorem (Theorem A.41,page 581), it has a convergent subsequence. But any Cauchy sequence which has aconvergent subsequence is itself convergent (its limit being that of the subsequence),see Exercise 1.6 on page 43. Hence any Cauchy sequence in is convergent.

Second case: Considering as a normed real vector space of dimension 2, we cansuppose that the base field is .

Consider a basis B= (b1, . . . , bd) of the vector space E. Then we deduce that E iscomplete from the case of the real numbers and the following two facts: (1) a sequence(xn)n of vectors, with coordinates (x1n , . . . , xdn ) in B, converges in E if and only ifeach coordinate sequence (xk n)n; and (2), if a sequence is a Cauchy sequence, theneach coordinate is a Cauchy sequence.

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This is simply un = 10n

10n

where is the integral part function.


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Sequences 15

Both facts can be checked immediately when the norm of E is defined byx =max|xk |, and other norms reduce to this case since all norms are equivalent on E.

Example 1.15 The space L2 of square integrable functions (in the sense of

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