Connection of the B¨ottcher equation with the parametrized Poisson integral

Abstract

In the paper, one of the real generalizations of the B¨ottcher equation is considered. It is shown that in some situations, after finding the particular solution of the generalized equation, it is possible to obtain other solutions of it. As an example, we describe a threeparameter family of real functional equations for a function of two arguments, for which particular solutions are found. This generalization has a wide field of application. Many quantities after a properly introduced parametrization satisfy the generalized B¨ottcher equation as a function of the parameters. As an illustration, we give two-parameter families generated by the determinant of a linear combination of second-order matrices. It is shown that the parametrized Poisson integral, as a function of its parameters, satisfies the generalized B¨ottcher equation. This allowed us to calculate the Poisson integral and the Euler integral in a new way. As a supplement, the calculation of the Poisson integral by the method of integral sums is presented.