CONCERNING ∫_{0}^{1} ⋯ ∫_{0}^{1}(x_{1}^{2} + ⋯ + x_{k}^{2})^{1/2}dx_{1} … dx_{k}

AND A TAYLOR SERIES METHOD

R. S. ANDERSSEN, R. P. BRENT, D. J. DALEY, AND P. A. P. MORAN

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} ≃ (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

Only the Abstract is given here. The full paper appeared as [1]. The first three values of m_{k} are

m_{1} = 1/2,
m_{2} = (\sqrt{2} + ln(1 + \sqrt{2})) /3 ≃ 0.76519572,
m_{3} = \sqrt{3}/4 + ln ((1 + \sqrt{3})/\sqrt{2}) − π/24 ≃ 0.96059196.

References

[1] R. S. Anderssen, R. P. Brent, D. J. Daley and P. A. P. Moran, “Concerning ∫_{0}^{1} ⋯ ∫_{0}^{1}(x_{1}^{2}+⋯+x_{k}^{2})^{1/2}dx_{1} … dx_{k} and a Taylor series method”, SIAM J. Applied Mathematics 30 (1976), 22–30. MR 52#15773, Zbl 337.65022.

(Anderssen, Brent) Computer Centre, Australian National University, Canberra

(Daley, Moran) Statistics Department, Australian National University, Canberra

1991 Mathematics Subject Classification. Primary 60D05; Secondary 60-04, 65-04, 65D30.

Key words and phrases. Uniform distribution, mean distance, Taylor series, asymptotic series, geometric probability, Cauchy-Schwarz inequality.

Received by the editors June 13, 1974.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1976, SIAM..

Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb033a typeset using AMS-L^{A}TEX.