A140133 Decimal expansion of the area enclosed in the lens-shaped region of the Laplace Limit.

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

Offset: 1

Author

Jonathan Vos Post, Jun 04 2008

Keywords

• cons
• nonn

Comments

See Weisstein for complex analysis function.

Examples

			1.8532684487079870332219364034397278879469653896325464...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..3000
• Eric W. Weisstein, Laplace Limit (value given is incorrect)

Crossrefs

Cf. A033259, A085984.

Programs

Mathematica

f[x_] := (Sqrt[x - Tanh[x]]*(x*Csch[x]^2 + 2*x - Coth[x]))/(2* Sqrt[-x + Coth[x]]); xmax = x /. FindRoot[Coth[x] - x == 0, {x, 1}, WorkingPrecision -> 200]; First[ RealDigits[ Chop[ Quiet[ NIntegrate[f[x], {x, 0, xmax}, WorkingPrecision -> 200, MaxRecursion -> 20]]*4], 10, 100]] (* Jean-François Alcover, Jun 07 2012, after D. S. McNeil *)

Sage

def A140133_cons(dps=200):
    from mpmath import mp, sqrt, tanh, coth, csch, findroot, quad
    mp.dps = 2*dps # safety
    def f(x): return 1/2*sqrt(x - tanh(x))*(x*csch(x)^2 + 2*x - coth(x))/sqrt(-x + coth(x))
    xmax = findroot(lambda x: coth(x)-x, 1)
    return quad(f, [0, xmax])*4  # D. S. McNeil, Feb 01 2011