Note on the reduction of dual and triple series equations to dual and triple integral equations

Abstract

Dual integral equations involving Bessel functions occur in the solution of some boundary-value problems in potential theory with conditions prescribed on a circular disc and a considerable amount of attention has been given to the solution of such equations (cf. (1)). The method of solving these dual integral equations is very similar to that employed in the solution of certain dual series equations involving Legendre functions. Equations of this type occur in problems in potential theory with conditions prescribed on a spherical cap and their solution has been obtained by Collins (2). No definite mathematical connexion has, however, been established between these dual series and dual integral equations and the object of this note is to establish such a connexion.