On the Laplace series

Abstract

The solution to one of the fundamental problems of mathematical physics, namely the Dirichlet problem for a sphere, is reduced, as is known, to the question of expanding a so-called “arbitrary” function of two angles into a series arranged according to the spherical Laplace's functions. The possibility of expansion for a function that has two first derivatives has been proven, and by reasoning similar to that presented in our article: “On the theory of trigonometric series”, it can be established that the expansion is also possible for a function that satisfies the Lipchitz’s condition.

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