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A262701 Decimal expansion of lowest Dirichlet eigenvalue of the Laplacian within a certain L-shaped region. 2


%S 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,

%T 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,

%U 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

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

%C 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.

%H Robert Stephen Jones, <a href="/A262701/b262701.txt">Table of n, a(n) for n = 1..1001</a>

%H Paolo Amore, John P. Boyd, Francisco M. Fernandez, Boris Rösler, <a href="http://arxiv.org/abs/1509.02795">High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson Extrapolation of second order finite differences</a>, arXiv:1509.02795 [physics.comp-ph], 2015.

%H Timo Betcke and Lloyd N. Trefethen, <a href="http://dx.doi.org/10.1137/S0036144503437336">Reviving the method of particular solutions</a>, SIAM Rev., 47 (2004), 469-491; <a href="http://eprints.maths.ox.ac.uk/1196/1/NA-03-12.pdf">Oxford, eprints.maths.ox.ac.uk, pdf</a>.

%H L. Fox, P. Henrici, and C. Moler, <a href="http://www.jstor.org/stable/2949737">Approximations and bounds for eigenvalues of elliptic operators</a>, SIAM J. Numer. Anal., 4 (1967), 89-102; <a href="http://blogs.mathworks.com/images/cleve/Fox_Henrici_Moler.pdf">blogs.mathworks.com, eprint, pdf</a>

%H Robert S. Jones, <a href="http://arxiv.org/abs/1602.08636">Computing ultra-precise eigenvalues of the Laplacian within polygons</a>, arXiv preprint arXiv:1602.08636, 2016

%H MathWorks, <a href="http://www.mathworks.com/company/newsletters/articles/the-mathworks-logo-is-an-eigenfunction-of-the-wave-equation.html">The mathworks logo is an eigenfunction of the wave equation</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eigenvalue">Dirichlet eigenvalue</a>

%e 9.639723844021941052711459262364823156267289525821906456109579700564035647863370...

%K nonn,cons

%O 1,1

%A _Robert Stephen Jones_, Sep 27 2015

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