数学物理

A book's apoLogy

Index of notation

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 theary 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 -algebra

2.2.e Negligible sets

2.2.f Lebesgue measure on Rn

2.2.g Definition ofthe Lebesgue integral

2.2.h Functions zero almost everywhere, space L1

2.2.1 And today?

Exercises

Solutions

3 Integral calculus

3.1 Integrability in practice

3.1.a Standard functions

3.l.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

4.2 Cauchy's theorem

4.3 Properties of holomorphic functions

4.4 Singularities of a function

4.5 Laurent series

……

5 Complex Analysis Ⅱ

6 Conformal maps

7 Distributions Ⅰ

8 Distributions II

9 Hilbert spaces, Fourier series

10 Fourier transform of functions

11 Fourier transform of distributions

12 The Laplace transform

13 Physical applications of the Fourier transform

14 Bras, kets, and all that sort of thing

15 Green functions

16 Tensors

17 Differential forms

18 Groups and group representations

19 Introduction to probability theory

20 Random variables

21 Convergence of random variables： central limit theorem

Appendices

Tables

There is a fairly fasbionable current of thought that bolds that the use of advanced mathematics is of little real use in physics， and goes sometimes as far as to say that knowing convinced that matbematics is stikll a parecious source of insight， not for students of physics， but also for researchers.
　　Many only see mathematics as a tool-and of course， it is part a tool， but they should be reminded that， as Galileo said， the book of Nature is written in give examples that knowing mathematics provides the means to understand precise physical notions， to use them more easily， to establish them on a sure foundation， and even more importantly， to discover new ones.