This book provides a unified introduction to special functions of mathematical physics, with applications in various fields. It is translated from the Russian by Ralph P. Boas. The authors are from the M.V. Keldish Institute of Applied Mathematics of the Academy of Sciences of the USSR. The book was originally published in Russian in 1978 and later translated into English in 1988. It includes a detailed table of contents, a preface, a foreword, and an introduction. The book covers the theory of special functions, including classical orthogonal polynomials, spherical harmonics, Bessel functions, and hypergeometric functions. It also discusses their applications in quantum mechanics, mathematical physics, and numerical analysis. The book is organized into chapters, each discussing specific types of special functions and their properties. The authors have developed a method based on a generalization of the Rodrigues formula for classical orthogonal polynomials, which allows for the derivation of explicit integral representations and basic properties of special functions. The book also includes a detailed discussion of the connection between special functions and quantum mechanics, as well as their applications in various physical problems. The authors have also included a section on the theory of special functions in the context of difference equations and non-uniform lattices. The book is intended for students and researchers in mathematical physics, quantum mechanics, and numerical analysis. It includes a list of references, an index of notations, and a list of figures. The authors have also provided a detailed explanation of the method used to study special functions, which is based on a generalization of the Rodrigues formula. The book is written in a clear and concise manner, with a focus on the essential properties and applications of special functions. The authors have also included a section on the connection between special functions and the theory of angular momentum, as well as their applications in the study of atomic spectra and scattering theory. The book is a comprehensive resource for understanding the theory and applications of special functions in mathematical physics and related fields.This book provides a unified introduction to special functions of mathematical physics, with applications in various fields. It is translated from the Russian by Ralph P. Boas. The authors are from the M.V. Keldish Institute of Applied Mathematics of the Academy of Sciences of the USSR. The book was originally published in Russian in 1978 and later translated into English in 1988. It includes a detailed table of contents, a preface, a foreword, and an introduction. The book covers the theory of special functions, including classical orthogonal polynomials, spherical harmonics, Bessel functions, and hypergeometric functions. It also discusses their applications in quantum mechanics, mathematical physics, and numerical analysis. The book is organized into chapters, each discussing specific types of special functions and their properties. The authors have developed a method based on a generalization of the Rodrigues formula for classical orthogonal polynomials, which allows for the derivation of explicit integral representations and basic properties of special functions. The book also includes a detailed discussion of the connection between special functions and quantum mechanics, as well as their applications in various physical problems. The authors have also included a section on the theory of special functions in the context of difference equations and non-uniform lattices. The book is intended for students and researchers in mathematical physics, quantum mechanics, and numerical analysis. It includes a list of references, an index of notations, and a list of figures. The authors have also provided a detailed explanation of the method used to study special functions, which is based on a generalization of the Rodrigues formula. The book is written in a clear and concise manner, with a focus on the essential properties and applications of special functions. The authors have also included a section on the connection between special functions and the theory of angular momentum, as well as their applications in the study of atomic spectra and scattering theory. The book is a comprehensive resource for understanding the theory and applications of special functions in mathematical physics and related fields.