Derivation of Specific Solutions and Asymptotic Analysis for the Cylindrical Dirichlet Problem

In this paper, we deduce a new formula that explicitly expresses the cylindrical Poisson kernel of any degree and order using cylindrical Poisson polynomials. We then demonstrate how this formula can be applied to solve the Dirichlet boundary value problems for cylindrical Laplace equations. Furthermore, we examine and discuss the behaviors of this solution approach at infinity.