AN INTEGRAL-EQUATION SOLUTION TO THE DIRICHLET PROBLEM FOR LAPLACES-EQUATION IN AN ELLIPSE

If u(z) = Re F(z) is the solution to the Dirichlet problem for Laplace's equation in an ellipse, then the analytic function F satisfies a symmetric integral equation on the interfocal segment. In this article, the iterated kernels are evaluated explicitly, and the characteristic functions and values of the associated homogeneous equation are found. These are, respectively, the Chebyshev polynomials Tn(z) (n = 0, 1, 2, ...) and powers of [ (a + b) (a - b)]2, where a and b are the semi-axes of the ellipse (a > b > 0). Two representations are given for the solution to the boundary value problem. The first is a series of Chebyshev polynomials, found earlier by Henrici. The second is analogous to the Poisson integral representation for the case of a circular boundary; it involves the theta function I1. © 1990.