The fundamental Laplacian eigenvalue of the ellipse with Dirichlet boundary conditions

TL;DR

This paper extends analytic expansions for the lowest Dirichlet eigenvalue of the Laplacian in an ellipse, using high-precision computations and polynomial fitting to improve understanding of eigenvalue behavior across eccentricities.

Contribution

It introduces a novel approach combining high-precision numerical data with polynomial fitting and integer relation algorithms to extend existing eigenvalue expansions.

Findings

Extended series near the circle with nine additional terms

Extended asymptotic expansion near the strip with four additional terms

Independent confirmation and extension of previous results

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.