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Box Integral

A box integral for dimension n with parameters q and s is defined as the expectation of distance from a fixed point q of a point r chosen at random over the unit n-cube,

X_n(s,q) =int_0^1...int_0^1_()_(n)[(r_1-q_1)^2+...+(r_n-q_n)^2]^(s/2)dr_1...dr_n
(1)

(Bailey et al. 2006).

Two special cases include

B_n(s)=int_0^1...int_0^1_()_(n)(r_1^2+...+r_n^2)^(s/2)dr_1...dr_n
(2)
Delta_n(s)=int_0^1...int_0^1_()_(2n)[(r_1-q_1)^2+...+(r_n-q_n)^2]^(s/2)dr_1...×dr_ndq_1...dq_n
(3)

which, with s=1, correspond to hypercube point picking (to a fixed vertex) and hypercube line picking, respectively.

Hypercube point picking to the center is given by

Z_n(s,(1/2,...,1/2)_()_(n))=(B_n(s))/(2^s).
(4)

See also

Hypercube Line Picking, Hypercube Point Picking, Unit Cube, Unit Square, Unit Square Integral

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References

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." Preprint. Apr. 3, 2006.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 238 and 272, 2007.

Referenced on Wolfram|Alpha

Box Integral

Cite this as:

Weisstein, Eric W. "Box Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoxIntegral.html

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