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A140133
Decimal expansion of the area enclosed in the lens-shaped region of the Laplace Limit.
0
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
1,2
COMMENTS
See Weisstein for complex analysis function.
LINKS
Eric W. Weisstein, Laplace Limit (value given is incorrect)
EXAMPLE
1.853268...
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 *)
PROG
(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]
CROSSREFS
Sequence in context: A116397 A275306 A176705 * A086723 A011406 A201488
KEYWORD
cons,nonn
AUTHOR
Jonathan Vos Post, Jun 04 2008
STATUS
approved