A352769 Decimal expansion of Pi^2 * log(2).

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

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

Table of n, a(n) for n=1..105.

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

Cf. A000796 (Pi), A002162 (log(2)), A002117 (zeta(3)), A002388 (Pi^2), A086054 (Pi*log(2)).

Sequence in context: A272488 A100608 A350795 * A335005 A321075 A234846

Adjacent sequences: A352766 A352767 A352768 * A352770 A352771 A352772

Keyword

nonn,cons

Author

Amiram Eldar, Apr 02 2022

Status

approved