*(English)*Zbl 1065.33001

The book consists of two parts. The first part contains the presentation of the foundation of the theory of Bessel functions, the second part contains the applications.

The first part is divided into two chapters. In the first chapter, the main properties of solutions of the homogeneous Bessel functions, which are based on the representation of the solutions as series in increasing powers of the argument, are considered. Then, the Bessel integral, the Poisson integral and some of their generalizations are presented. The Neumann addition theorems and the foundations of the theory of products of Bessel functions are considered. Questions concerning differential equations which can be reduced to the Bessel equation and the inhomogeneous Bessel equations are discussed in detail. In particular, functions contiguous to the Bessel functions are considered.

The second chapter contains the theory of definite and improper integrals as well as elements of the theory of dual integral equations. The representation of functions by the series of Fourier-Bessel, Dini, and Schlömilch is presented. Consideration of problems concerning homogeneous Bessel equations which are more complicated than those considered in the first chapter leads to Lommel functions in two variables. Partial cylindrical functions and asymptotic expansions of Bessel functions are discussed.

The second part is also devoted to applications and consists of two chapters. In the first chapter, problems about plates and shells of rotation including problems on the oscillations of a circular plate and the equilibrium of a plate on a Winkler-type foundation are considered, as well as problems which can be solved by the method of compensating loadings. The method of initial parameters is considered in detail in the framework of the problem of axially symmetrical deformation of a circular conic shell.

In the second chapter, comparatively different problems of oscillation theory, hydrodynamics, heat theory, etc. are collected. In Section 11, the boundary-value problems whose solution can be reduced to singular equations are considered. Formally, the author considers plates and membranes; however, the solution has a more general nature and is less connected with concrete applications.