1 - Basic definitions and propositions

Abstract

This chapter introduces necessary notations, basic facts, and other propaedeutic information. It starts with definitions and most useful properties of special functions: gamma, beta, Bessel, and hypergeometric functions. After that functional spaces are listed which are used in the book. Further integral transforms are introduced, including different ones with special function kernels. The Mellin transform is essential for this book, especially coupled with Slater's theorem. Some multi-dimensional transforms are listed. At the end basic facts on singular differential equations are mentioned, including those with Bessel operators, and for them an important classification of I. A. Kipriyanov is formulated. Basic facts on Tricomi and Euler–Poisson–Darboux equations are introduced.