The fundamental Laplacian eigenvalue of the ellipse with Dirichlet boundary conditions

Abstract: In this project, I examine the lowest Dirichlet eigenvalue of the Laplacian within the ellipse as a function of eccentricity. Two existing analytic expansions of the eigenvalue are extended: Close to the circle (eccentricity near zero) nine terms are added to the Maclaurin series; and near the infinite strip (eccentricity near unity) four terms are added to the asymptotic expansion. In the past, other methods, such as boundary variation techniques, have been used to work on this problem, but I use a different approach -- which not only offers independent confirmation of existing results, but extends them. My starting point is a high precision computation of the eigenvalue for selected values of eccentricity. These data are then fit to polynomials in appropriate parameters yielding high-precision coefficients that are fed into an LLL integer-relation algorithm with forms guided by prior results.