A254980 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(-1) in Bailey's paper).
9, 6, 7, 4, 1, 2, 0, 2, 1, 2, 4, 1, 1, 6, 5, 8, 9, 8, 6, 6, 1, 8, 3, 6, 4, 3, 8, 1, 7, 8, 1, 5, 8, 3, 9, 0, 1, 3, 5, 9, 3, 7, 0, 0, 9, 2, 9, 9, 9, 6, 0, 7, 0, 7, 2, 7, 4, 8, 2, 5, 7, 9, 2, 6, 6, 9, 5, 2, 4, 8, 4, 1, 9, 6, 7, 2, 3, 8, 4, 0, 5, 6, 6, 7, 2, 3, 1, 0, 2, 5, 3, 2, 3, 4, 2, 7, 7, 0, 0, 6, 6, 6, 6, 9
Offset: 0
Examples
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Links
- D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
- D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2.
- Eric Weisstein's MathWorld, Inverse Tangent Integral.
- Eric Weisstein's MathWorld, Polylogarithm.
- Eric Weisstein's MathWorld, Box Integral.
Programs
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Mathematica
Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[-1] = 2*Log[3] - (2/3) * Catalan + 2*Ti2[3 - 2*Sqrt[2]] - Sqrt[8]*ArcTan[1/Sqrt[8]] // Re; RealDigits[ B4[-1], 10, 104] // First
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Python
from mpmath import * mp.dps=105 x=3 - 2*sqrt(2) Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x)) C = 2*log(3) - (2/3)*catalan + 2*Ti2x - sqrt(8) * atan(1/sqrt(8)) print([int(n) for n in list(str(C.real)[2:-1])]) # Indranil Ghosh, Jul 03 2017
Formula
B_4(-1) = 2*log(3) - (2/3)*Catalan + 2*Ti_2(3-2*sqrt(2)) - sqrt(8) * arctan( 1/sqrt(8) ), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).
Extensions
Name corrected by Amiram Eldar, Jun 04 2023