|
%I #28 Dec 23 2018 14:38:57
%S 2,5,2,0,0,9,7,3,7,9,2,9,3,2,4,6,4,6,7,6,0,6,5,2,1,2,2,3,9,5,3,8,5,4,
%T 7,7,0,2,8,7,8,0,6,5,3,2,2,5,5,6,6,1,4,6,4,9,7,9,0,1,5,3,9,4,4,7,7,3,
%U 6,0,5,4,2,4,0,2,9,8,2,8,3,6,7,4,5,6,6,2,0,7,3,7,1,3,4,1,5,7,8,5
%N Decimal expansion of C[12] coefficient in 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the Pi-area, N-sided regular polygon.
%C This is the 12th coefficient C[12] in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of Pi-area, N-sided regular polygon. It was determined using experimental mathematics by computing the coefficient to 125 digits of precision. It can be computed using the expression in the Formula section. It is expressed in terms of L0 = [A115368]^2 = [A244355] = 5.78318... (eigenvalue of unit-radius circle) and Riemann zeta functions. Although this is derived using experimental mathematics, the decimal expansion reported is equal to that expression. In context, the eigenvalue expression for the N-sided, Pi-area regular polygon is
%C L = L0*(1 + C[3]/N^3 + C[5]/N^5 + C[6]/N^6 + C[7]/N^7 + C[8]/N^8 + ... + C[12]/N^12 + ...). The expression for this coefficient follows a pattern similar to lower-order coefficients (except C[11] [A321215]), e.g., C[3]=4*zeta(3) and C[5]=(12-2*L0)*zeta(5).
%H Robert Stephen Jones, <a href="/A321216/b321216.txt">Table of n, a(n) for n = 5..1004</a>
%H Mark Boady, <a href="http://hdl.handle.net/1860/idea:6852">Applications of Symbolic Computation to the Calculus of Moving Surfaces</a>. PhD thesis, Drexel University, Philadelphia, PA. 2015.
%H P. Grinfeld and G. Strang, <a href="https://doi.org/10.1016/j.jmaa.2011.06.035">Laplace eigenvalues on regular polygons: A series in 1/N</a>, J. Math. Anal. Appl., 385-149, 2012.
%H Robert Stephen Jones, <a href="https://arxiv.org/abs/1712.06082">The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions</a>, arXiv:1712.06082 [math.NA], 2017.
%H Robert Stephen Jones, <a href="https://doi.org/10.1007/s10444-017-9527-y">Computing ultra-precise eigenvalues of the Laplacian within polygons</a>, Advances in Computational Mathematics, May 2017.
%e 25200.9737929324646760652122395385477028780653225566146497901539447736054240...
%o (PARI) {default(realprecision,100);L0=solve(x=2,3,besselj(0,x))^2;(32/3+272*L0/3-16*L0^2)*zeta(3)^4+(1360/3-488*L0/3+456*L0^2+107*L0^3/3+5*L0^4/8)*zeta(3)*zeta(9)+(432-216*L0-207*L0^2+47*L0^3/2+11*L0^4/8)*zeta(5)*zeta(7)}
%Y Cf. A321215 is decimal expansion of C[11], the next lower order coefficient.
%Y Cf. A115368, A244355, A002117, and A013663.
%K nonn,cons
%O 5,1
%A _Robert Stephen Jones_, Oct 31 2018
|
