OFFSET
0,1
LINKS
Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.
Maurice Horowitz, Probability of random paths across elementary geometrical shapes, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; Correction, ibid., Vol. 3, No. 1 (1966), p. 285.
FORMULA
Equals (1/(3*Pi)) * (2*Pi - 6 + 2*log(2) + 7*log(3)/2 + 4*sqrt(2)*arccot(sqrt(2))) - (4/Pi) * Integral_{x=1..sqrt(2)} (sqrt(x^2-1) * (x * arccot(x) + log(1 + x^2)/2) / x) dx.
EXAMPLE
0.5977557435927337398151960798274735969724820222495278516...
MATHEMATICA
RealDigits[N[(1/(3*Pi)) * (2*Pi - 6 + 2*Log[2] + 7*Log[3]/2 + 4*Sqrt[2]*ArcCot[Sqrt[2]]) - (4/Pi) * Integrate[Sqrt[x^2-1] * (x * ArcCot[x] + Log[1 + x^2]/2) / x, {x, 1, Sqrt[2]}], 110], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 29 2021
STATUS
approved