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A160439 Decimal expansion of a constant that appears in flux/diffusion problems with trapping surfaces. 0
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, 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, 3, 0, 1, 4, 8, 3, 8, 6, 6, 5, 5, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The constant appears as a correction in effective radii of flux problems of particles undergoing certain random walks in one or three dimensions.
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.)
LINKS
E. G. Coffman, P. Flajolet, L. Flato, and M. Hofri, The maximum of random walk and its application to rectangle packing, Probability in Engineering and Informational Sciences 12:373-386 (1998).
S. N. Majumdar, A. Comtet, and R. M. Ziff, Unified solution of the expected maximum of a discrete time random walk and the discrete flux to a spherical trap, J. Stat. Phys. 122 (2006), 833-856.
Robert M. Ziff, Flux to a trap, J. Stat. Phys. 65 (1991), 1217-1233.
FORMULA
Equals (-1/Pi) * Integral_{x=0..oo} log( (6/x^2)*(1-sin(x)/x) ) / x^2 dx.
EXAMPLE
0.29795219028004776416465987228031204613834651480951717550256...
MAPLE
evalf(-1/Pi * Int(log(6/x^2*(1-sin(x)/x))/x^2, x=0..infinity), 20); # Vaclav Kotesovec, Mar 17 2015
MATHEMATICA
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]]
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 *)
CROSSREFS
Sequence in context: A241753 A157350 A121837 * A245287 A254595 A137453
KEYWORD
cons,nonn
AUTHOR
Robert M. Ziff, May 13 2009
EXTENSIONS
Definition condensed by R. J. Mathar, May 30 2009
Corrected decimal places 39-46 and added more decimals by Vaclav Kotesovec, Mar 18 2015
More terms from Vaclav Kotesovec, Dec 07 2016
STATUS
approved

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Last modified February 24 14:26 EST 2024. Contains 370305 sequences. (Running on oeis4.)