cp's OEIS Frontend

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

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).

This is the 11th coefficient C[11] = -6016.337... in the 1/N expansion of the lowest Laplacian Dirichlet eigenvalue of the N-sided, Pi-area regular polygon.

In context, the eigenvalue expression for the N-sided, Pi-area regular polygon is L = L0*(1 + C[3]/N^3 + C[5]/N^5 + C[6]/N^6 + C[7]/N^7 + C[8]/N^8 + ... + C[11]/N^11 + ...) where L0 = [A115368]^2 = [A244355] is the eigenvalue in the Pi-area circle.

C[11] was computed by first computing several hundred 200-digit eigenvalues in the range from N = 1000 to 3000, and then using linear regression to determine this expansion coefficient. All digits reported are correct. This is the first coefficient that appears to break the regular pattern involving roots of Bessel functions and Riemann zeta functions, for example, C[3] = 4*zeta(3) and C[5] = (12-2*L0)*zeta(5), where zeta(n) is the Riemann zeta function. C[11] is negative.

The eigenvalues of the Laplacian within the regular pentagon with Dirichlet boundary conditions are calculated to at least 1000 digits. The ratio of the second eigenvalue to the first is calculated and expressed as a continued fraction. The ratio has an advantage since it is independent of the pentagon area. All terms in this expansion are correct.

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.

This is the lowest Dirichlet eigenvalue of the Helmholtz equation within the unit-edged regular pentagon. It has been calculated (RSJ, Sep 2015) to just over 1500 decimal digits.