OFFSET
1,1
COMMENTS
Rainer and Serene (1976) used the sum that is given in the first formula in the calculation of the free energy of superfluid Helium-3. They evaluated the sum by 6.8.
Rainwater (1978) found the integral representation of this sum, which is given in the second formula, and evaluated it by 6.84109 +- 0.00001.
Glasser and Ruehr (1981) proved the sum is equal to this constant.
REFERENCES
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, Philadelphia, 1987, pp. 187-188.
LINKS
D. Bierens de Haan, Nouvelles tables d'intégrales définies, Leide, 1867, Table 206, eq. 10.
M. L. Glasser, An Infinite Triple Summation, Problem 80-13, SIAM Review, Vol. 22, No. 3 (1980), pp. 363-364; Solutions by the proposer and by O. G. Ruehr, ibid., Vol. 23, No. 3 (1981), pp. 393-394.
C. F. Lindman, Examen des nouvelles tables d'intégrales définies de m. Bierens de Haan, Amsterdam 1867, Stockholm, 1891, p. 99, Tab. 206, 10.
D. Rainer and J. W. Serene, Free energy of superfluid He 3, Phys. Rev. B, Vol. 13, No. 11 (1976), pp. 4745-4765; Erratum, ibid., Vol. 18, No. 7 (1978), p. 3760.
J. C. Rainwater, Evaluation of frequency sums for the free energy of superfluid He 3, Phys. Rev. B, Vol. 18, No. 7 (1978), pp. 3728-3729.
FORMULA
Equals Sum_{i,j,k, positive and negative odd integers} sign(i) * sign(j) * sign(k) * sign(i+j-k)/(i^2*j^2).
Equals -8 * Integral_{x=0..1} arctanh(x)*log(x)/(x*(1-x^2)) dx - 7*zeta(3)/2.
Equals Integral_{x=0..Pi/2} (4*x^2*cos(x) - x*(Pi-x))/sin(x) dx (Bierens de Haan, 1867; Lindman, 1891).
EXAMPLE
6.84108846385711654484747915395409607129977904818791...
MATHEMATICA
RealDigits[Pi^2*Log[2], 10, 100][[1]]
PROG
(PARI) Pi^2 * log(2) \\ Michel Marcus, Apr 02 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Apr 02 2022
STATUS
approved