  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.
