The subject of this work is the study of functions that occur initially in the form of a definite integral (see article). As a simple example to explain the application of the obtained formulas, I give the classical problem of the motion of a spherical pendulum in the case of its small oscillations around the lowest equilibrium position. Examples of problems leading to the generalized Kepler's equation. The origin of Bessel's functions of many variables and their expression in the form of infinite series (see article). The author pays attention to equations satisfied by generalized Bessel's functions and the general solution of these equations.
