A244354 -id:A244354 - cp's OEIS frontend

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

Jean-François Alcover, Jun 26 2014

			5.7831859629467845211759957584558...

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

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

Cf. A115368, A244354.

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

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

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

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.

A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2f'(x)^2 dx <= Kintegral_{0..1} f'(x)^4 dx.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 05 2014

			0.34611896560593345099609054206876591598395281385974864...

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

David W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. Volume 30, Number 2 (1969), 367-383
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 30.

Cf. A242440, A244263, A244347, A244354, A245292.

(* Using Boyd's formula *) I0[p_, q_, r_] := Integrate[(((q - 1)*t + 1)*t^(1/p - 1))/ (((r*(q - 1))*t)/(r - q) + 1)^((r*p + p + q)/(r*p)), {t, 0, 1}]; K[p_, q_, r_] := (beta = (((r - 1)*p + (r - q))/((r - 1)*(p + q)))^(1/r); (((r - q)*p^p)*beta^(p + q - r))/(I0[p, q, r]^p*((r - 1)*(p + q)))); RealDigits[K[2, 2, 4], 10, 104] // First

24/(2*sqrt(3) + 3*sqrt(2)*arcsinh(sqrt(2)))^2.

A248914 Decimal expansion of L = Integral_{t=0..1} 1/(1-2t^2/3) dt, an auxiliary constant associated with one of the integral inequalities studied by David Boyd.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 16 2014

			1.40382196515535516573036373889961027748003532830665702207...

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

David W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. Volume 30, Number 2 (1969), 367-383
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 31.

Cf. A242440, A244263, A244347, A244354, A245292, A246859.

RealDigits[Sqrt[3/2]*ArcTanh[Sqrt[2/3]], 10, 101] // First

L = sqrt(3/2)*arctanh(sqrt(2/3)).

K = A246859 = 2/(L+1)^2.

cp's OEIS Frontend

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.

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A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2f'(x)^2 dx <= Kintegral_{0..1} f'(x)^4 dx.

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A248914 Decimal expansion of L = Integral_{t=0..1} 1/(1-2t^2/3) dt, an auxiliary constant associated with one of the integral inequalities studied by David Boyd.

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cp's OEIS Frontend

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.

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PARI

PARI

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A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2*f'(x)^2 dx <= K*integral_{0..1} f'(x)^4 dx.

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A248914 Decimal expansion of L = Integral_{t=0..1} 1/(1-2t^2/3) dt, an auxiliary constant associated with one of the integral inequalities studied by David Boyd.

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A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2f'(x)^2 dx <= Kintegral_{0..1} f'(x)^4 dx.