On the first Laplacian eigenvalue of an elliptical membrane

This paper introduces a refined approach for approximating the first Dirichlet Laplacian eigenvalue of elliptical domains using a combination of Gaussian and Student-like functions. We present a precise formula that achieves an error margin of less than \(0.4\%\) $0.4\%$ and derive improved theoretical bounds for the eigenvalue. The results include new inequalities and insights into the asymptotic behavior of the approximation formula. This work enhances the accuracy of eigenvalue predictions for ellipses, with implications for applied mathematics, physics, and engineering.