1 Reminders: convergence of sequences and series

1.1 The problem of limits in physics

1.1.a Two paradoxes involving kinetic energy

1.1.b Romeo, Juliet, and viscous fluids

1.1.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

2 Measure theory and the Lebesgue integral

2.1 The integral according to Mr. Riemann

2.1.a 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

2.2.b Borel subsets

2.2.c Lebesgue measure

2.2.d The Lebesgue σ-algebra

2.2.e Negligible sets

2.2.f Lebesgue measure on ？

2.2.g Definition of the Lebesgue integral

2.2.h Functions zero almost everywhere, space L1

2.2.i And today?

Exercises

Solutions

3 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

4 Complex Analysis Ⅰ

4.1 Holomorphic functions

4.1.a Definitions

4.1.b Examples

4.1.c The operators ？/？z and ？/？？

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

4.6 Applications to the computation of horrifying integrals or ghastly sums

4.6.a Jordan's lemmas

4.6.b Integrals on ？ of a rational function

4.6.c Fourier integrals

4.6.d Integral on the unit circle of a rational function

4.6.e Computation of infinite sums

Exercises

Problem

Solutions

5 Complex Analysis Ⅱ

5.1 Complex logarithm; multivalued functions

5.1.a 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 f 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

6 Conformal maps

6.1 Conformal maps

6.1.a 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

7 Distributions Ⅰ

7.1 Physical approach

7.1.a The problem of distribution of charge

7.1.b The problem of momentum and forces during an elastic shock

7.2 Definitions and examples of distributions

7.2.a Regular distributions

7.2.b Singular distributions

7.2.c Support of a distribution

7.2.d Other examples

7.3 Elementary properties. Operations

7.3.a Operations on distributions

7.3.b Derivative of a distribution

7.4 Dirac and its derivatives

7.4.a The Heaviside distribution

7.4.b Multidimensional Dirac distributions

7.4.c The distribution δ′

7.4.d Composition of δ with a function

7.4.e Charge and current densities

7.5 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 ？

7.5.c Laplacian of a function discontinuous along a surface ？

7.5.d Application: laplacian of 1/r in 3-space

7.6 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

7.7 Physical interpretation of convolution operators

7.8 Discrete convolution

8 Distributions Ⅱ

8.1 Cauchy principal value

8.1.a 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 ？

8.2.b Sequences of functions converging to δ

8.2.c Convergence in ？ 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

9 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 e2 space

9.3.c The space L2 [0,a]

9.3.d The L2(？) 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 f ∈ 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

10.1.a Definition

10.1.b Examples

10.1.c The L1 space

10.1.d Elementary properties

10.1.e Inversion

10.1.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 ？

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

11.1.a Tempered distributions

11.1.b Fourier transform of tempered distributions

11.1.c Examples

11.1.d Higher-dimensional Fourier transforms

11.1.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

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

13 Physical applications of the Fourier transform

13.1 Justification of sinusoidal regime analysis

13.2 Fourier transform of vector fields: longitudinal and transverse fields

13.3 Heisenberg uncertainty relations

13.4 Analytic signals

13.5 Autocorrelation of a finite energy function

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

14.1.a 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 ？> ∈ H

14.2.b Bras <？ ∈ H′

14.2.c Generalized bras

14.2.d Generalized kets

14.2.e Id = ∑n ？n> <？n？

14.2.f Generalized basis

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

15.5 Quantum mechanics

15.6 Klein-Gordon equation

Exercises

16 Tensors

16.1 Tensors in affine space

16.1.a Vectors

16.1.b Einstein convention

16.1.c Linear forms

16.1.d Linear maps

16.1.e Lorentz transformations

16.2 Tensor product of vector spaces: tensors

16.2.a Existence of the tensor product of two vector spaces

16.2.b Tensor product of linear forms: tensors of type (0 2)

16.2.c Tensor product of vectors: tensors of type (2 0)

16.2.d Tensor product of a vector and a linear form: linear maps or (1 1)-tensors

16.2.e Tensors of type (p q)

16.3 The metric, or, how to raise and lower indices

16.3.a Metric and pseudo-metric

16.3.b Natural duality by means of the metric

16.3.c Gymnastics: raising and lowering indices

16.4 Operations on tensors

16.5 Change of coordinates

16.5.a Curvilinear coordinates

16.5.b Basis vectors

16.5.c Transformation of physical quantities

16.5.d Transformation of linear forms

16.5.e Transformation of an arbitrary tensor field

16.5.f Conclusion

Solutions

17 Differential forms

17.1 Exterior algebra

17.1.a 1-forms

17.1.b Exterior 2-forms

17.1.c Exterior k-forms

17.1.d Exterior product

17.2 Differential forms on a vector space

17.2.a Definition

17.2.b Exterior derivative

17.3 Integration of differential forms

17.4 Poincaré's theorem

17.5 Relations with vector calculus: gradient, divergence, curl

17.5.a Differential forms in dimension 3

17.5.b Existence of the scalar electrostatic potential

17.5.c Existence of the vector potential

17.5.d Magnetic monopoles

17.6 Electromagnetism in the language of differential forms 480

Problem

Solution

18 Groups and group representations

18.1 Groups

18.2 Linear representations of groups

18.3 Vectors and the group SO(3)

18.4 The group SU(2) and spinors

18.5 Spin and Riemann sphere

Exercises

19 Introduction to probability theory

19.1 Introduction

19.2 Basic definitions

19.3 Poincaré formula

19.4 Conditional probability

19.5 Independent events

20 Random variables

20.1 Random variables and probability distributions

20.2 Distribution function and probability density

20.2.a Discrete random variables

20.2.b (Absolutely) continuous random variables

20.3 Expectation and variance

20.3.a Case of a discrete r.v.

20.3.b Case of a continuous r.v.

20.4 An example: the Poisson distribution

20.4.a Particles in a confined gas

20.4.b Radioactive decay

20.5 Moments of a random variable

20.6 Random vectors

20.6.a Pair of random variables

20.6.b Independent random variables

20.6.c Random vectors

20.7 Image measures

20.7.a Case of a single random variable

20.7.b Case of a random vector

20.8 Expectation and characteristic function

20.8.a Expectation of a function of random variables

20.8.b Moments, variance

20.8.c Characteristic function

20.8.d Generating function

20.9 Sum and product of random variables

20.9.a Sum of random variables

20.9.b Product of random variables

20.9.c Example: Poisson distribution

20.10 Bienaymé-Tchebychev inequality

20.10.a Statement

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

Appendices

A Reminders concerning topology and normed vector spaces

A.1 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

B Elementary reminders of differential calculus

B.1 Differential of a real-valued function

B.1.a Functions of one real variable

B.1.b Differential of a function f : ？n → ？

B.1.c Tensor notation

B.2 Differential of map with values in ？p

B.3 Lagrange multipliers

Solution

C Matrices

C.1 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

D A few proofs

Tables

Fourier transforms

Laplace transforms

Probability laws

Further reading

References

Portraits

Sidebars

Index