A355415 Decimal expansion of the average distance between the center of a unit cube to a point on its surface uniformly chosen by a random direction from the center.
6, 1, 0, 6, 8, 7, 4, 0, 1, 9, 5, 1, 5, 8, 3, 8, 5, 6, 5, 3, 4, 6, 6, 7, 2, 2, 9, 6, 7, 3, 7, 1, 6, 6, 2, 8, 4, 6, 9, 1, 1, 5, 5, 2, 5, 8, 1, 9, 0, 7, 4, 6, 2, 7, 5, 8, 9, 9, 2, 9, 9, 4, 1, 0, 2, 5, 9, 6, 8, 1, 5, 7, 3, 6, 2, 8, 8, 6, 6, 4, 1, 3, 7, 2, 1, 4, 5, 0, 5, 5, 9, 6, 5, 7, 6, 6, 0, 8, 0, 8, 3, 3, 5, 7, 2
Offset: 0
Examples
0.61068740195158385653466722967371662846911552581907...
Links
- Mathematics Stack Exchange, Average Width of the Cube, or Average Radius of a Regular Polyhedron, 2021.
- Mathematics Stack Exchange, What is the average radius of the hypercube?, 2019.
Crossrefs
Programs
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Mathematica
RealDigits[N[(3/Pi)*Integrate[ArcCot[Sqrt[1 + x^2]]/Sqrt[1 + x^2], {x, 0, 1}], 101], 10, 100][[1]] (* or *) RealDigits[3 * ((Im[PolyLog[2, (3 - 2*Sqrt[2])*I]] - Catalan)/Pi - Log[17 - 12*Sqrt[2]]/8), 10, 100][[1]]
Formula
Equals (1/2) * Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-1) dx dy / Integral_{x=-1..1, y=-1..1} (1 + x^2 + y^2)^(-3/2) dx dy.
Equals (3/Pi) * Integral_{x=0..1} arccot(sqrt(1+x^2))/sqrt(1+x^2) dx.
Equals (6/Pi) * Integral_{x=0..Pi/4} log(sqrt(1+cos(x)^2)/cos(x)) dx.
Equals 3 * ((Im(Li_2((3-2*sqrt(2))*i)) - G)/Pi - log(17-12*sqrt(2))/8), where Li_2 is the dilogarithm function, i is the imaginary unit, and G is Catalan's constant (A006752).
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