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A033259
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Decimal expansion of Laplace's limit constant.
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12
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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EXAMPLE
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0.662743419349181580974742097109252907056233549115022417520392534990971853086...
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MATHEMATICA
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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 *)
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PROG
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(PARI) sqrt(solve(u=1, 2, tanh(u)-1/u)^2-1) \\ M. F. Hasler, Feb 01 2011
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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