A033259 Decimal expansion of Laplace's limit constant.

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

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• cons

Comments

Maximum value taken by the function x/cosh(x), which occurs at A085984. - Hrothgar, Mar 12 2014

Given two equal coaxial circular rings of diameter D located in two parallel planes distant d apart, this constant is the maximum value of d / D so that there exists a catenoid resting on these two rings. - Robert FERREOL, Feb 07 2019

The maximum value of the eccentricity for which the Lagrange series expansion for the solution to Kepler's equation converges. Laplace (1827) calculated the value 0.66195. The Italian astronomer Francesco Carlini (1783 - 1862) found the limit 0.66 five years before Laplace (Sacchetti, 2020). - Amiram Eldar, Aug 17 2020

Examples

			0.662743419349181580974742097109252907056233549115022417520392534990971853086...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.
• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 402.
• John Oprea, The Mathematics of Soap Films: Explorations with Maple, Amer. Math. Soc., 2000, p. 183.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..20000
• Steven R. Finch, Laplace Limit Constant [Broken link]
• Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
• J. J. Green, The Lipschitz constant for the radial projection on real l_p - implementation notes, 2012. - _N. J. A. Sloane_, Sep 19 2012
• Pierre-Simon Laplace, Supplément au 5e volume du Traité de mécanique céleste, Paris (1827). See p. 11.
• Simon Plouffe, The laplace limit constant(to 500 digits)
• Andrea Sacchetti, Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations, Historia Mathematica (2020), preprint, arXiv:2002.02679 [math.HO], 2020.
• Eric Weisstein's World of Mathematics, Laplace Limit.
• Eric Weisstein's World of Mathematics, Kepler's Equation.
• Wikipedia, Laplace limit.
• Index entries for transcendental numbers.

Crossrefs

Cf. A033260, A033261, A033262, A033263, A085984.

Programs

Mathematica

x/.FindRoot[ x Exp[ Sqrt[ 1+x^2 ] ]/(1+Sqrt[ 1+x^2 ])==1, {x, 1} ]
Sqrt[x^2 - 1] /. FindRoot[ x == Coth[x], {x, 1}, WorkingPrecision -> 30 ] (* Leo C. Stein, Jul 30 2017 *)
RealDigits[Sqrt[Root[{# - (1 + #)/E^(2 #) - 1 &, 1.1996786}]^2 - 1], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

PARI

sqrt(solve(u=1,2,tanh(u)-1/u)^2-1) \\ M. F. Hasler, Feb 01 2011

Formula

Equals sqrt(A085984^2-1). - Jean-François Alcover, May 14 2013