The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions

TL;DR

This paper investigates the lowest Laplacian eigenvalue of regular polygons with Dirichlet boundary conditions, providing numerical evidence, conjecturing additional terms in its asymptotic expansion, and analyzing convergence properties for large S.

Contribution

It offers independent numerical validation of previous analytical results, proposes an extended asymptotic expansion, and studies the series' convergence behavior for large polygon sides.

Findings

Numerical eigenvalues computed to fifty digits for polygons with up to 150 sides.

Conjecture of two additional terms in the asymptotic expansion.

Higher-order coefficients alternate in sign and grow rapidly, indicating divergence for small S.

Abstract

The lowest eigenvalue of the Laplacian within the S-sided regular polygon with Dirichlet boundary conditions is the focus of this report. As suggested by others, this eigenvalue may be expressed as an asymptotic expansion in powers of 1/S where, interestingly, they have shown that the first few coefficients in that expansion, up to sixth order, may be expressed analytically in terms of Riemann zeta functions and roots of Bessel functions. This report builds on that work with three main contributions: (1) compelling numerical evidence independently supporting those published results, (2) a conjecture adding two more terms to the asymptotic expansion, and (3) an observation that higher-order coefficients both alternate in sign and grow rapidly in magnitude, which suggest the series doesn't converge unless S>=10. This report is based on a numerical computation of the eigenvalues precise to fifty digits for S up to 150. Keywords: Laplacian eigenvalue; regular polygon; asymptotic expansion