## Concerning [an integral] and a Taylor series method

33. R. S. Anderssen, R. P. Brent, D. J. Daley and P. A. P. Moran,
Concerning
and a Taylor series method,
* SIAM J. Applied Mathematics* 30 (1976), 22-30.
MR 52#15773.
Abstract:
dvi (3K),
pdf (81K).

Paper:
pdf (648K).

## Abstract

The integral of the title equals the mean distance
*m*_{k} from the origin
of a point uniformly distributed over the *k*-dimensional
unit hypercube I^{k}.
Closed form expressions are given for
*k* = 1, 2 and 3,
while for general *k*,
*m*_{k} is asymptotic to
(*k*/3)^{1/2}.
Using * inter alia * methods from geometry,
Cauchy-Schwarz inequalities
and Taylor series expansions, several inequalities and an asymptotic series
for *m*_{k} are established.
The Taylor series method also yields a slowly
converging infinite series for *m*_{k}
and can be applied to more general
problems including the mean distance between two points independently
distributed at random in I^{k}.

## Comments

For recent references, see the MathWorld
"Hypercube Line Picking" page.
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