A076414 -id:A076414 - cp's OEIS frontend

A244347 Decimal expansion of 'mu', a Sobolev isoperimetric constant related to the "rod inequality", arising from the elasticity study of a rod that is clamped at both ends.

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

Offset: 0

Jean-François Alcover, Jun 26 2014

			0.001997746934053886262...

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

Cf. A076414 (theta).

digits = 104; theta = x /. FindRoot[Cos[x]*Cosh[x] == 1, {x, 5}, WorkingPrecision -> digits+10]; mu = 1/theta^4; Join[{0, 0}, RealDigits[mu, 10, digits] // First]

mu = 1/theta^4, where theta is A076414.

A076415 Decimal expansion of second solution of equation cos(x) cosh(x) = 1.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Oct 10 2002

This is an equation related to a beam clumped at both ends: cos(x) cosh(x) = 1. The first three solutions are: 4.73 (A076414), 7.853 (this sequence) and 10.996 (A076416).

			cos(x) cosh(x) = 1, x = 7.8532...

Z. Guede, I. Elishakov, A fifth-order polynomial that serves as both.., Chaos, Solitons and Fractals 12 (2001) 1267-1298.

Cf. A076414, A076416.

RealDigits[x/.FindRoot[Cos[x] Cosh[x]==1,{x,5 Pi/2},WorkingPrecision->120]][[1]] (* Jean-Francois Alcover, Mar 14 2011 *)

A076416 Decimal expansion of third solution of equation cos(x)*cosh(x) = 1.

Original entry on oeis.org

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

Offset: 2

Zak Seidov, Oct 10 2002

This is an equation related to a beam clumped at both ends: cos(x) cosh(x) = 1. The first three solutions are: 4.73 (A076414), 7.853 (A076415) and 10.9956 (this sequence).

			10.995607838001670906669...

Z. Guede and I. Elishakov, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos, Solitons and Fractals 12 (7) (2001) 1267-1298.

Cf. A076414, A076415.

RealDigits[x/.FindRoot[Cos[x] Cosh[x]==1,{x,7 Pi/2},WorkingPrecision -> 120]][[1]] (* Jean-François Alcover, Mar 14 2011 *)

solve(x=9,11,cos(x)*cosh(x)-1) \\ Charles R Greathouse IV, Mar 15 2011

Offset corrected by R. J. Mathar, Feb 05 2009

A244350 Decimal expansion of 'lambda', a Sobolev isoperimetric constant related to the "rod inequality", arising from the elasticity study of a rod that is clamped at both ends.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 26 2014

			5.13878013260283423689422...

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

Cf. A076414 (theta), A244347 (mu).

digits = 103; theta = x /. FindRoot[Cos[x]*Cosh[x] == 1, {x, 5}, WorkingPrecision -> digits+10]; lambda = theta^4/Pi^4; RealDigits[lambda, 10, digits] // First

lambda = theta^4/Pi^4 = 1/(Pi^4*mu), where theta is A076414 and mu is A244347.

lambda is also the smallest eigenvalue of the ODE g''''(x)=lambda*g(x), g(0)=g'(0)=g(Pi)=g'(Pi)=0.

cp's OEIS Frontend

A244347 Decimal expansion of 'mu', a Sobolev isoperimetric constant related to the "rod inequality", arising from the elasticity study of a rod that is clamped at both ends.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A076415 Decimal expansion of second solution of equation cos(x) cosh(x) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A076416 Decimal expansion of third solution of equation cos(x)*cosh(x) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A244350 Decimal expansion of 'lambda', a Sobolev isoperimetric constant related to the "rod inequality", arising from the elasticity study of a rod that is clamped at both ends.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula