R. COURANT and D. HILBERT, Methods of Mathematical Physics, GTTI, Moscow–Leningrad, 1933, 525 pp., price 8 rubles 50 kopecks.

Interest in the problems of quantum mechanics and mathematical physics, which has grown in recent years among our physicists and mathematicians, makes the appearance of R. Courant and D. Hilbert’s book extremely timely. It would be no exaggeration to say that without it an in-depth study of quantum theory is impossible, and that it is necessary for anyone who intends to work creatively in the field of theoretical physics. The book is an excellent textbook and, in this capacity, can be recommended. Anyone wishing to study, for example, integral equations must turn to the specialized works of Lovitt or Villat published by GTTI. Nor can one expect to learn variational calculus from it. But what makes the book especially valuable is that it gives the reader the opportunity to obtain a fairly accurate idea of contemporary problems and methods of mathematical physics. Problems are not considered separately from one another, but in their mutual connection; methods are not limited merely to applications to particular problems, but possess that degree of generality which is needed in order to be able to apply them when the conditions of the problem encountered by the physicist have their own special features.

Only quantum physics has shown how close are the methods of algebraic analysis and the theory of functions. Heisenberg’s quantum mechanics and Schrödinger’s wave mechanics differ only in their mathematical formulation. Therefore the path chosen by Courant and Hilbert is especially close to the physicist: the transition from vector algebra to the theory of expansions of functions in a row by orthogonal functions.

The first chapter, using vector algebra as an example, formulates the problem of expansion in a series of functions and the problem of integral equations, to which the next two chapters are devoted. The fourth chapter treats the fundamental problems and methods of the calculus of variations. The fifth and sixth chapters are devoted to the central problem of eigenvalues and its connection with integral equations and the calculus of variations. The last chapter contains an exposition of the theory of Bessel and Legendre functions and concludes with the theory of asymptotic expansions.

Whoever has mastered the contents of the book, whoever has succeeded in finding the interconnection of all the diversity of the material, will be fully equipped to solve the problems and tasks advanced before the physicist by the new quantum mechanics.

It is useful to cite Courant’s own opinion. He does not recommend that the beginner read the book consecutively and systematically. It is better to begin the book at several places at once. The ideas and methods are so intertwined that, with such reading, they complement one another, and what is not readily understood in one place may become clear when another is read. This advice can only be recommended.

Yu. Rumer