Madelung Constants
The quantities obtained from cubic, hexagonal, etc., Lattice Sums, evaluated at 
, are calledMadelung constants. For cubic Lattice Sums, they are expressible in closed form for Even indices,
 |  |  | (1) |
 | |  | (2) |
is given by Benson's Formula, | (3) |
is sometimes called ``the'' Madelung constant,corresponds to the Madelung constant for a 3-D NaCl crystal, and is numerically equal to 
.For hexagonal Lattice Sum, 
is expressible in closed form as
 | (4) |
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Buhler, J. and Wagon, S. ``Secrets of the Madelung Constant.'' Mathematica in Education and Research 5, 49-55, Spring 1996. Crandall, R. E. and Buhler, J. P. ``Elementary Function Expansions for Madelung Constants.'' J. Phys. Ser. A: Math. and Gen. 20, 5497-5510, 1987. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mdlung/mdlung.html
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