A033259 -id:A033259 - cp's OEIS frontend

A248916 Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace's limit constant A033259.

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

Offset: 1

Jean-François Alcover, Oct 16 2014

The boundary value problem y''(x) + c*exp(y(x)) = 0, y(0) = y(1) = 0 and c > 0, has 0, 1 or 2 solutions when c > gamma, c = gamma and c < gamma, respectively. [After Steven Finch]

			3.5138307191251612062078370932388235871...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.8 Laplace limit constant, p. 266.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 32.
Eric Weisstein's MathWorld, Laplace Limit

Cf. A033259, A085984.

digits = 104; lambda = x /. FindRoot[x Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> digits + 5]; gamma = 8*lambda^2; RealDigits[gamma, 10, digits] // First

A345736 Decimal expansion of the surface area of the catenoid-shaped soap film between two parallel coaxial unit-radius circular rings whose distance between their centers is maximal (2*A033259).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 25 2021

			7.53780320580457797882651822069283034551193592152611...

G. C. Greubel, Table of n, a(n) for n = 1..20000
Markus Deserno, Notes on Differential Geometry, with special emphasis on surfaces in R^3, 2004. See p. 28.
Lev A. Slobozhanin, J. Iwan, D. Alexander and Viral D. Patel, The stability margin for stable weightless liquid bridges, Physics of Fluids, Vol. 14, No. 1 (2002), pp. 209-224. See p. 222.
Eric Weisstein's World of Mathematics, Minimal Surface of Revolution.

Cf. A033259, A085984, A253545.

RealDigits[2 * Pi * x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120], 10, 100][[1]]

Equals 2*Pi*A085984.

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 06 2003

This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - Carl R. White, Dec 09 2003. Also the root of x*tanh x = 1. - N. J. A. Sloane, May 07 2020
This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - Michael Clausen, Feb 18 2013
This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - Jean-François Alcover, Sep 08 2014, after Steven Finch.
For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - Bernard Schott, Mar 13 2020

			1.1996786402577338339163698486411419442614587884186072...

O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
Index entries for transcendental numbers.

Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
  (* Aaron Hendrickson, Mar 17 2021 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - Aaron Hendrickson, Mar 17 2022

A033260 Continued fraction for Laplace's limit constant.

Original entry on oeis.org

0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, 2, 4, 1, 5, 1, 1, 2, 1601, 2, 63, 1, 2, 12, 17, 9, 2, 1, 1, 57, 1, 6, 2, 4, 3, 10, 1, 6, 2, 1, 3, 4, 2, 1, 31, 3, 6, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 14, 1, 4, 1, 2, 2, 1, 97, 1, 52, 2, 4, 1, 1, 5, 13, 1, 1, 3, 6, 2, 3, 3, 2, 2, 1, 1, 2, 28, 2, 3, 1, 4, 2, 1, 4

Offset: 0

Eric W. Weisstein

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.

G. C. Greubel, Table of n, a(n) for n = 0..20000
S. R. Finch, Laplace Limit Constant
Eric Weisstein's World of Mathematics, Laplace Limit

Cf. A033259 (decimal expansion), A033261, A033262, A033263.

ContinuedFraction[Sqrt[x^2-1]/.FindRoot[x==Coth[x], {x,1}, WorkingPrecision -> 250], 121] (* G. C. Greubel, Dec 13 2024 *)

Offset changed by Andrew Howroyd, Jul 04 2024

A033261 Position of first occurrence of n in the continued fraction for the Laplace's limit constant.

Original entry on oeis.org

1, 9, 34, 12, 14, 31, 100, 8, 25, 35, 101, 23, 72, 57, 750, 270, 24, 365, 363, 482, 191, 642, 821, 541, 393, 632, 4, 85, 2049, 617, 44, 201, 941, 182, 206, 862, 3104, 1295, 2122, 258, 1576, 5551, 158, 3353, 3870, 114, 506, 1669, 9646, 1127, 445, 66, 1804

Offset: 1

Eric W. Weisstein

nonn

The continued fraction expansion is indexed [a_0; a_1, a_2, a_3, ...].

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

Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Laplace Limit.

Cf. A033259, A033260, A033262, A033263.

A033260(a(n)) = n. - Andrew Howroyd, Sep 12 2024

More terms from Michel ten Voorde Jun 20 2003

Terms decreased by 1 for consistency with offset change in A033260 by Andrew Howroyd, Sep 12 2024

A033262 Incrementally largest terms in the continued fraction for Laplace's limit constant.

Original entry on oeis.org

1, 27, 154, 1601, 2135, 4132, 7166, 8391, 10970, 34639, 748645, 1005174

Offset: 1

Eric W. Weisstein

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

Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Laplace Limit.

Cf. A033259, A033260, A033261, A033263.

More terms from Michel ten Voorde Jun 20 2003

A033263 Positions of the incrementally largest terms in the continued fraction for Laplace's limit constant.

Original entry on oeis.org

1, 4, 10, 18, 1800, 2079, 6560, 7688, 11310, 12437, 24708, 63577

Offset: 1

Eric W. Weisstein

The continued fraction expansion is indexed [a_0; a_1, a_2, a_3, ...].

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

Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Laplace Limit.

Cf. A033259, A033260, A033261, A033262.

A033260(a(n)) = A033262(n). - Andrew Howroyd, Sep 12 2024

More terms from Michel ten Voorde Jun 20 2003

Terms decreased by 1 for consistency with offset change in A033260 by Andrew Howroyd, Sep 12 2024

A240358 Decimal expansion of 'c', a constant linked to an estimate of density of zeros of an entire function of exponential type.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 04 2014

			1.508879561538319928909884488160578573694278589...

Alexandre Eremenko and Peter Yuditskii, An extremal problem for a class of entire functions

Cf. A033259.

FindRoot[Log[c + Sqrt[c^2 + 1]] == Sqrt[1 + 1/c^2], {c, 3/2}, WorkingPrecision -> 100][[1, 2]] // RealDigits[#, 10, 100]& // First

Solution to log(c + sqrt(c^2 + 1)) = sqrt(1 + 1/c^2).

Equals 1/A033259. - Robert FERREOL, Jun 16 2025

A249136 Decimal expansion of the largest constant 'beta' for which there exists a solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 22 2014

			1.874521464034264187600324820470264120147219398917056...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024, p. 32.
Eric Weisstein's MathWorld, Laplace Limit.

Cf. A033259, A085984, A248916.

lambda = x /. FindRoot[x*Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> 102]; beta = Sqrt[8]*lambda; RealDigits[beta] // First

beta = sqrt(8)*lambda, where lambda is A033259, the Laplace limit constant 0.66274...

Equals sqrt(A248916). - Hugo Pfoertner, Dec 19 2024

A253545 Decimal expansion of r = 0.527697..., a boundary ratio separating catenoid and Goldschmidt solutions in the minimal surface of revolution problem.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 21 2015

Consider two circular frames each of diameter D and with a separation of d.
If d/D < r = 0.527697..., then a catenoid gives the absolute minimum area.
If r < d/D < L = 0.66274... (Laplace limit), there are 3 minimal surfaces of revolution passing through the frames: 2 catenoids and the so-called Goldschmidt discontinuous solution consisting of the 2 disks.
If d/D > L, there remains only the Goldschmidt solution.

			0.5276973969625715285724233433631805779688537906314195417222751595...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Robert Ferréol's MathCurve, Catenoid
Eric Weisstein's MathWorld, Laplace Limit
Eric Weisstein's MathWorld, Minimal Surface of Revolution

Cf. A033259 (Laplace limit), A202357.

digits = 105; u0 = u /. FindRoot[u*Sqrt[u^2-1] + ArcCosh[u] - u^2 == 0, {u, 6/5}, WorkingPrecision -> digits+5];  r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First

arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^2-1) + arccosh(u) - u^2 = 0.
Solution of 2*cosh((x^2+1)/2) = x+1/x. - Robert FERREOL, Feb 07 2019
Equals sqrt(A202357). - Hugo Pfoertner, Dec 21 2024

cp's OEIS Frontend

A248916 Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace's limit constant A033259.

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A345736 Decimal expansion of the surface area of the catenoid-shaped soap film between two parallel coaxial unit-radius circular rings whose distance between their centers is maximal (2*A033259).

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A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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A033260 Continued fraction for Laplace's limit constant.

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A033261 Position of first occurrence of n in the continued fraction for the Laplace's limit constant.

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A033262 Incrementally largest terms in the continued fraction for Laplace's limit constant.

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A033263 Positions of the incrementally largest terms in the continued fraction for Laplace's limit constant.

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A240358 Decimal expansion of 'c', a constant linked to an estimate of density of zeros of an entire function of exponential type.

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