A262701 Decimal expansion of lowest Dirichlet eigenvalue of the Laplacian within a certain L-shaped region.

9, 6, 3, 9, 7, 2, 3, 8, 4, 4, 0, 2, 1, 9, 4, 1, 0, 5, 2, 7, 1, 1, 4, 5, 9, 2, 6, 2, 3, 6, 4, 8, 2, 3, 1, 5, 6, 2, 6, 7, 2, 8, 9, 5, 2, 5, 8, 2, 1, 9, 0, 6, 4, 5, 6, 1, 0, 9, 5, 7, 9, 7, 0, 0, 5, 6, 4, 0, 3, 5, 6, 4, 7, 8, 6, 3, 3, 7, 0, 3, 9, 0, 7, 2, 2, 8, 7, 3, 1, 6, 5, 0, 0, 8, 7, 9, 6, 7, 8, 8, 8, 3, 1, 1, 5

Offset: 1

Author

Robert Stephen Jones, Sep 27 2015

Keywords

• nonn
• cons

Comments

This is the lowest Dirichlet eigenvalue of the Laplacian within the famous L-shape formed by joining two unit-edged squares to adjacent edges of a third. The familiar logo of MathWorks, publisher of MATLAB, is created from the corresponding lowest eigenfunction, with some artistic license. I simply extended the original Fox-Henrici-Moler 1967 eigenvalue calculation to just over 1000 digits using a method substantially identical to the method described in the Fox et al. paper.

Examples

			9.639723844021941052711459262364823156267289525821906456109579700564035647863370...
		

Links

• Robert Stephen Jones, Table of n, a(n) for n = 1..1001
• Paolo Amore, John P. Boyd, Francisco M. Fernandez, Boris Rösler, High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences, arXiv:1509.02795 [physics.comp-ph], 2015.
• Timo Betcke and Lloyd N. Trefethen, Reviving the method of particular solutions, SIAM Rev., 47 (2004), 469-491; Oxford, eprints.maths.ox.ac.uk, pdf.
• L. Fox, P. Henrici, and C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), 89-102; blogs.mathworks.com, eprint, pdf
• Robert S. Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, arXiv preprint arXiv:1602.08636, 2016
• MathWorks, The mathworks logo is an eigenfunction of the wave equation
• Wikipedia, Dirichlet eigenvalue