A185576 Decimal expansion of Born's basic potential Pi_0.

2, 8, 3, 7, 2, 9, 7, 4, 7, 9, 4, 8, 0, 6, 1, 9, 4, 7, 6, 6, 6, 5, 9, 1, 7, 1, 0, 4, 6, 0, 7, 7, 3, 8, 8, 2, 2, 3, 8, 9, 2, 1, 8, 7, 0, 2, 1, 5, 8, 4, 8, 3, 5, 9, 9, 0, 0, 3, 7, 1, 9, 0, 0, 6, 9, 9, 9, 2, 4, 7, 7, 1, 1, 1, 6, 2, 2, 7, 3, 3, 0, 9, 4, 7, 4, 0, 4, 1, 5, 3, 0, 7, 9, 2, 7, 1, 1, 0, 3, 5

Offset: 1

Author

R. J. Mathar, Jan 31 2011

Keywords

• cons
• nonn

Comments

Decimal expansion of Sum'_{m,n,p = -infinity..infinity} 1/(m^2 + n^2 + p^2)^s, analytic continuation to s=1/2. The prime at the sum symbol means the term at m=n=p=0 is omitted.

Examples

			2.8372974794806194766659171046...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.

Links

• Y. Sakamoto, Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures, J. Chem. Phys. 28 (1958) 164, variable Pi_0.
• I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable a(1).
• I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable a(1).

Crossrefs

Cf. A185577, A185578, A185579, A185580, A185581, A185582, A185583.

Programs

Mathematica

digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n)) * Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], Print["s(", k, ") = ", s[k]]; k = k + dk]; RealDigits[s[k], 10, digits] // First (* Jean-François Alcover, Sep 10 2014 *)

Formula

Equals A085469/3 + A185577 + A185578.

Extensions

More terms from Jean-François Alcover, Sep 10 2014