A140133 - OEIS

%I #16 Feb 10 2014 01:28:43

%S 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,

%T 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,

%U 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

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

%C See Weisstein for complex analysis function.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/LaplaceLimit.html">Laplace Limit</a> (value given is incorrect)

%e 1.853268...

%t 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_ *)

%o (Sage)

%o def A140133_cons(dps=200):

%o     from mpmath import mp, sqrt, tanh, coth, csch, findroot, quad

%o     mp.dps = 2*dps # safety

%o     def f(x): return 1/2*sqrt(x - tanh(x))*(x*csch(x)^2 + 2*x - coth(x))/sqrt(-x + coth(x))

%o     xmax = findroot(lambda x: coth(x)-x, 1)

%o     return quad(f, [0, xmax])*4  # [_D. S. McNeil_, Feb 01 2011]

%Y Cf. A033259, A085984.

%K cons,nonn

%O 1,2

%A _Jonathan Vos Post_, Jun 04 2008