A273989 Decimal expansion of the odd Bessel moment t(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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

Offset: 0

Author

Jean-François Alcover, Jun 06 2016

Keywords

• cons
• nonn

Examples

			0.660344869018672357837266831705994263854241991696873858300803587553894...
		

Links

• David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Crossrefs

Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)), A273986 (s(5,5)), A273990 (t(5,3)), A273991 (t(5,5)).

Programs

Mathematica

t[5, 1] = NIntegrate[x*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 105]; RealDigits[t[5, 1]][[1]]
(* or: *)
t[5, 1] = 4(7 - 4*Sqrt[3]) EllipticK[1 - 32/(16 + 7*Sqrt[3] - Sqrt[15])] EllipticK[1 - 32/(16 + 7*Sqrt[3] + Sqrt[15])]; RealDigits[t[5, 1], 10, 105][[1]]
RealDigits[EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32] EllipticK[(16 - 7 Sqrt[3] + Sqrt[15])/32]/4, 10, 105][[1]] (* Jan Mangaldan, Jan 06 2017 *)

Formula

Integral_{0..inf} x*BesselI_0(x)^2*BesselK_0(x)^3.

Equals 4(7 - 4*sqrt(3)) EllipticK(1 - 32/(16 + 7*sqrt(3) - sqrt(15))) EllipticK(1 - 32/(16 + 7*sqrt(3) + sqrt(15))).