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Benson's Formula

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An equation for a lattice sum b_3(1) (Borwein and Bailey 2003, p. 26)

b_3(1)=sum^'_(i,j,k=-infty)^infty((-1)^(i+j+k))/(sqrt(i^2+j^2+k^2))
(1)
=-12pisum_(m,n=1,3,...)^(infty)sech^2(1/2pisqrt(m^2+n^2)).
(2)

Here, the prime denotes that summation over (0, 0, 0) is excluded. The sum is numerically equal to -1.74756... (OEIS A085469), a value known as "the" Madelung constant.

No closed form for b_3(1) is known (Bailey et al. 2006).

See also

Lattice Sum, Madelung Constants

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References

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. §4.3.2 and 4.3.3 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 301, 1987.Finch, S. R. "Madelung's Constant." §1.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 76-81, 2003.Sloane, N. J. A. Sequence A085469 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Benson's Formula

Cite this as:

Weisstein, Eric W. "Benson's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BensonsFormula.html

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