A160439 - OEIS

%I #31 Dec 14 2024 09:17:56

%S 2,9,7,9,5,2,1,9,0,2,8,0,0,4,7,7,6,4,1,6,4,6,5,9,8,7,2,2,8,0,3,1,2,0,

%T 4,6,1,3,8,3,4,6,5,1,4,8,0,9,5,1,7,1,7,5,5,0,2,5,6,8,1,5,1,8,5,9,4,0,

%U 3,0,1,4,8,3,8,6,6,5,5,2

%N Decimal expansion of a constant that appears in flux/diffusion problems with trapping surfaces.

%C The constant appears as a correction in effective radii of flux problems of particles undergoing certain random walks in one or three dimensions.

%C Also related to correction term to the asymptotics of sums of random numbers uniformly distributed on an interval (see Coffman et al., who also present a double-sum formula.)

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 327.

%H E. G. Coffman, P. Flajolet, L. Flato, and M. Hofri, <a href="https://www.researchgate.net/publication/29600601_The_Maximum_of_a_Random_Walk_and_Its_Application_to_Rectangle_Packing">The maximum of random walk and its application to rectangle packing</a>, Probability in Engineering and Informational Sciences 12:373-386 (1998).

%H S. N. Majumdar, A. Comtet, and R. M. Ziff, <a href="http://dx.doi.org/10.1007/s10955-005-9002-x">Unified solution of the expected maximum of a discrete time random walk and the discrete flux to a spherical trap</a>, J. Stat. Phys. 122 (2006), 833-856.

%H Robert M. Ziff, <a href="http://dx.doi.org/10.1007/BF01049608">Flux to a trap</a>, J. Stat. Phys. 65 (1991), 1217-1233.

%F Equals (-1/Pi) * Integral_{x=0..oo} log( (6/x^2)*(1-sin(x)/x) ) / x^2 dx.

%e 0.29795219028004776416465987228031204613834651480951717550256...

%p evalf(-1/Pi * Int(log(6/x^2*(1-sin(x)/x))/x^2, x=0..infinity),20); # _Vaclav Kotesovec_, Mar 17 2015

%t For[i = 0; s = 0, i < 100, i++, s = s + -(1/Pi)NIntegrate[Log[(1 - Sin[x]/ x)/(x^2/6)]/x^2, {x, 2 i Pi, 2 (i + 1) Pi}, WorkingPrecision -> 100]; Print[s]]

%t RealDigits[-1/Pi * Integrate[Log[(6/x^2) * (1 - Sin[x]/x)]/x^2, {x, 0, Infinity}], 10, 100][[1]] (* _Alonso del Arte_, Mar 18 2015 *)

%K cons,nonn

%O 0,1

%A _Robert M. Ziff_, May 13 2009

%E Definition condensed by _R. J. Mathar_, May 30 2009

%E Corrected decimal places 39-46 and added more decimals by _Vaclav Kotesovec_, Mar 18 2015

%E More terms from _Vaclav Kotesovec_, Dec 07 2016