Mathematics for physics and physicists. Transl. from the French by Emmanuel Kowalski.

*(English)*Zbl 1128.00002
Princeton, NJ: Princeton University Press (ISBN 978-0-691-13102-3/hbk). xxiv, 642 p. (2007).

There is a law, if not a physical law, that ensures that whenever one is using a standard mathematical technique for a physical problem it is always the special case or a first principles argument that is required. Nothing is straightforward! For such a situation this book is ideal. It presents clear definitions and the rationale for such definitions. The breadth of material is too wide here to discuss it all in detail but I will highlight some of the things that particularly interested me, together with a survey of the scope of the topics covered. The style of the book is very readable and an interesting biographical asides of the mathematicians associated with the topics provide light relief from the depth of the the analysis.

It begins with a discussion of sequences and series ( power series and asymptotic series). The beginning of this section sets the “tone” for the rest of the book by introducing and discussing two paradoxes to illustrate the idea that double limits cannot necessarily interchanged. It then moves on to a discussion of the theory of integration and this is one of the simplest and yet rigorous treatments that I have seen on both Riemann and Lebesgue integration, measure and Borel sets. It then goes on to discuss what it calls “integrability in practice” and this leads to complex analysis. Most of this is standard but very well written with particularly clear definitions of Holomorphic and Harmonic functions. The theory of residues is examined and the definitions of multivalued functions, cuts and Riemann surfaces and winding numbers are given. Again the clarity is to be admired and there is a particularly interesting discussion the the saddle point method for asymptotic approximations to integrals.

One of the topics that interested me most of all was the Fourier analysis and the theory of distributions. The theory of distributions is introduced initially from a physical point of view before the full rigor is given. The references to the initiators (Schwartz and Gelfand) are given but it is a pity the later work of Lighthill is omitted since this would have given a different perspective. There is a clear and detailed discussion of the principal value of integrals and Feynman’s modification of the standard theory and following this an introduction to Hilbert spaces and function spaces is given leading on to integral transforms, both in terms of ordinary functions and distributions. Physical applications are alluded to throughout. In particular the section on physical optics and the Dirac delta function properties is particularly illuminating. Laplace and Z-transforms are also discussed in these chapters.

In addition, as would be expected in a book of mathematical physics, the theory required for quantum mechanics is treated in detail. The uncertainty principle for position and momentum is presented as an exercise and the solution is also given. (Solutions are given for many of the exercises in the book). There is also a chapter on Green functions with applications to the heat equation, and quantum mechanics. These applications are really illustrating the way the mathematics is used and the subtleties of the theory rather than direct physical applications. This is followed by chapters on the theory of tensors and differential forms. The topic of symmetry is then examined with a chapter on groups, group representations and infinitesimal transformations. A particular Lie algebra (\(\text{SO}(3)\)) is examined in detail and the related \(\text{SU}(2)\) which is the basis for the theory of spinors.

In chapter 19 an introduction to probability theory is given. What is essentially a set of random numbers is given which is those numbers \(x\) for which the decimal expansion of \(x\) contains statistically as many zeros as ones, twos, threes, etc. The probability that a particular number belongs to this set with probability 1, but one cannot easily prove that any number has this property. This paradox is typical of those introduced to illustrate the concepts. Following this discussion the basic concepts of probability spaces, independent events, conditional probability and particularly Bayes theorem are introduced. The final two chapters deal with random variables and distributions, continuous and discrete, with an application to radioactive decay. The sum of random variables is examined and this leads to the development of the law of large numbers and to the central limit theorem.

There are several appendices which fill in details and give longer proofs omitted in the text. Thus the book is both a valuable reference book and is a good pedagogic treatment of mathematical physics. It is a book that should be on many bookshelves.

It begins with a discussion of sequences and series ( power series and asymptotic series). The beginning of this section sets the “tone” for the rest of the book by introducing and discussing two paradoxes to illustrate the idea that double limits cannot necessarily interchanged. It then moves on to a discussion of the theory of integration and this is one of the simplest and yet rigorous treatments that I have seen on both Riemann and Lebesgue integration, measure and Borel sets. It then goes on to discuss what it calls “integrability in practice” and this leads to complex analysis. Most of this is standard but very well written with particularly clear definitions of Holomorphic and Harmonic functions. The theory of residues is examined and the definitions of multivalued functions, cuts and Riemann surfaces and winding numbers are given. Again the clarity is to be admired and there is a particularly interesting discussion the the saddle point method for asymptotic approximations to integrals.

One of the topics that interested me most of all was the Fourier analysis and the theory of distributions. The theory of distributions is introduced initially from a physical point of view before the full rigor is given. The references to the initiators (Schwartz and Gelfand) are given but it is a pity the later work of Lighthill is omitted since this would have given a different perspective. There is a clear and detailed discussion of the principal value of integrals and Feynman’s modification of the standard theory and following this an introduction to Hilbert spaces and function spaces is given leading on to integral transforms, both in terms of ordinary functions and distributions. Physical applications are alluded to throughout. In particular the section on physical optics and the Dirac delta function properties is particularly illuminating. Laplace and Z-transforms are also discussed in these chapters.

In addition, as would be expected in a book of mathematical physics, the theory required for quantum mechanics is treated in detail. The uncertainty principle for position and momentum is presented as an exercise and the solution is also given. (Solutions are given for many of the exercises in the book). There is also a chapter on Green functions with applications to the heat equation, and quantum mechanics. These applications are really illustrating the way the mathematics is used and the subtleties of the theory rather than direct physical applications. This is followed by chapters on the theory of tensors and differential forms. The topic of symmetry is then examined with a chapter on groups, group representations and infinitesimal transformations. A particular Lie algebra (\(\text{SO}(3)\)) is examined in detail and the related \(\text{SU}(2)\) which is the basis for the theory of spinors.

In chapter 19 an introduction to probability theory is given. What is essentially a set of random numbers is given which is those numbers \(x\) for which the decimal expansion of \(x\) contains statistically as many zeros as ones, twos, threes, etc. The probability that a particular number belongs to this set with probability 1, but one cannot easily prove that any number has this property. This paradox is typical of those introduced to illustrate the concepts. Following this discussion the basic concepts of probability spaces, independent events, conditional probability and particularly Bayes theorem are introduced. The final two chapters deal with random variables and distributions, continuous and discrete, with an application to radioactive decay. The sum of random variables is examined and this leads to the development of the law of large numbers and to the central limit theorem.

There are several appendices which fill in details and give longer proofs omitted in the text. Thus the book is both a valuable reference book and is a good pedagogic treatment of mathematical physics. It is a book that should be on many bookshelves.

Reviewer: Brian L. Burrows (Stafford)

##### MSC:

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

00A05 | Mathematics in general |