A244355 Decimal expansion of 'lambda', a Sobolev isoperimetric constant related to the "membrane inequality", arising from the study of a vibrating membrane that is stretched across the unit disk and fastened at its boundary.

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

Offset: 1

Author

Jean-François Alcover, Jun 26 2014

Keywords

• nonn
• cons
• easy

Examples

			5.7831859629467845211759957584558...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants, p. 221.

Links

• Robert Stephen Jones, The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions, arXiv:1712.06082 [math.NA], 2017, p. 17.
• Eric Weisstein's MathWorld, Bessel Function Zeros
• Index entries for transcendental numbers

Crossrefs

Cf. A115368, A244354.

Programs

Mathematica

theta = BesselJZero[0, 1]; lambda = theta^2; RealDigits[lambda, 10, 103] // First

PARI

solve(x=2, 3, besselj(0, x))^2 \\ Michel Marcus, Nov 02 2018

PARI

besseljzero(0)^2 \\ Charles R Greathouse IV, Aug 09 2022

Formula

lambda = theta^2 where theta is A115368, the first positive zero of the Bessel function J0(x).

lambda = 1/mu = 1/A244354.

lambda is also the smallest eigenvalue of the ODE r^2*g''(r)+r*g'(r)+lambda*r^2*g(r)=0, g(0)=1, g(1)=0.