An improved method for dual trigonometrical series

Abstract

In a previous paper in this journal [1], I gave formulae for determining the coefficients in certain dual trigonometrical series. The derivation of these formulae involved rather sophisticated assumptions and some intricate manipulation of the hypergeometric function and relied heavily on the solution of Schlömilch's integral equation. I have now found a much simpler formal solution by using Mehler's integral representation of the Legendre polynomial and the final formulae for the coefficients can be given in a more attractive form. As the results of my previous work have had several recent applications to physical problems, it seems worth while to give some details of this improved solution.