Series equations involving Jacobi polynomials and mixed boundary-value problems of the Laplace equation

Abstract

A formal analysis of series equations involving Jacobi polynomials is given. ( 2 N + 1 ) \left ( {2N + 1} \right ) series equations involving Jacobi polynomials are reduced to a set of N N simultaneous Fredholm integral equations which can be solved numerically by the use of the Legendre-Gauss quadrature formula. In case of triple series equations the result is in agreement with that of Lowndes. Besides triple series equations, certain quadruple series equations can be also reduced to a single Fredholm integral equation of the second kind. Owing to the introduction of an arbitrary weight factor, the theory is feasible for the analysis of various many-part mixed boundary-value problems of the Laplace equation. As an example, special cases of certain trigonometric series equations are discussed in detail in connection with an electrostatic problem.