Construction of Green’s function of the Neumann problem in a ball

Representation of the Green’s function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that Green’s function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel was constructed for $n = 4$ and $n = 5$.