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A033259 Decimal expansion of Laplace's limit constant. 12

%I #55 Jul 15 2022 15:04:16

%S 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,

%T 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,

%U 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

%N Decimal expansion of Laplace's limit constant.

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

%C 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

%C 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

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.

%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 402.

%D John Oprea, The Mathematics of Soap Films: Explorations with Maple, Amer. Math. Soc., 2000, p. 183.

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/lpc/lpc.html">Laplace Limit Constant</a> [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010413222037/http://www.mathsoft.com/asolve/constant/lpc/lpc.html">Laplace Limit Constant</a> [From the Wayback machine]

%H J. J. Green, <a href="http://soliton.vm.bytemark.co.uk/pub/jjg/code/lcrp-inote.pdf">The Lipschitz constant for the radial projection on real l_p - implementation notes</a>, 2012. - _N. J. A. Sloane_, Sep 19 2012

%H Pierre-Simon Laplace, <a href="https://books.google.com/books?id=vjAbe1_wWUcC">Supplément au 5e volume du Traité de mécanique céleste, Paris (1827). See p. 11.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/laplace.txt">The laplace limit constant(to 500 digits)</a>

%H Andrea Sacchetti, <a href="https://doi.org/10.1016/j.hm.2020.06.001">Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations</a>, Historia Mathematica (2020), <a href="https://arxiv.org/abs/2002.02679">preprint</a>, arXiv:2002.02679 [math.HO], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KeplersEquation.html">Kepler's Equation</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Laplace_limit">Laplace limit</a>.

%F Equals sqrt(A085984^2-1). - _Jean-François Alcover_, May 14 2013

%e 0.662743419349181580974742097109252907056233549115022417520392534990971853086...

%t x/.FindRoot[ x Exp[ Sqrt[ 1+x^2 ] ]/(1+Sqrt[ 1+x^2 ])==1, {x, 1} ]

%t Sqrt[x^2 - 1] /. FindRoot[ x == Coth[x], {x, 1}, WorkingPrecision -> 30 ] (* _Leo C. Stein_, Jul 30 2017 *)

%t RealDigits[Sqrt[Root[{# - (1 + #)/E^(2 #) - 1 &, 1.1996786}]^2 - 1], 10, 100][[1]] (* _Eric W. Weisstein_, Jul 15 2022 *)

%o (PARI) sqrt(solve(u=1,2,tanh(u)-1/u)^2-1) \\ _M. F. Hasler_, Feb 01 2011

%Y Cf. A033259 - A033263, A085984.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)