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A273990
Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).
2
2, 5, 2, 2, 5, 3, 9, 4, 4, 8, 9, 7, 8, 4, 1, 9, 6, 5, 9, 9, 4, 5, 0, 9, 6, 1, 2, 5, 5, 5, 0, 9, 0, 4, 0, 8, 7, 7, 5, 0, 6, 8, 4, 5, 0, 7, 5, 5, 9, 7, 0, 0, 9, 9, 9, 2, 0, 6, 5, 9, 3, 0, 9, 4, 5, 2, 8, 9, 7, 1, 0, 2, 0, 7, 4, 1, 9, 8, 6, 0, 5, 9, 0, 8, 1, 5, 6, 3, 5, 4, 9, 5, 9, 6, 5, 1, 7, 4, 1, 1, 9, 2, 7, 9
OFFSET
0,1
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.
FORMULA
Integral_{0..inf} x*BesselI_0(x)^2*BesselK_0(x)^3.
EXAMPLE
0.252253944897841965994509612555090408775068450755970099920659309452897...
MATHEMATICA
t[5, 3] = NIntegrate[x^3*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]
(* or: *)
K[k_] := EllipticK[k^2/(-1+k^2)]/Sqrt[1-k^2];
D0[y_] := (4*y*K[Sqrt[((1-3*y)*(1+y)^3)/((1+3*y)*(1-y)^3)]])/Sqrt[(1+3*y)* (1-y)^3];
t[5, 3] = NIntegrate[4y^2*(1-2y^2+4y^4)*D0[y]/(1-4y^2)^(5/2), {y, 0, 1/3}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]
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)), A273989 (t(5,1)), A273991 (t(5,5)).
Sequence in context: A171937 A065291 A065267 * A216022 A100955 A049413
KEYWORD
nonn,cons
AUTHOR
STATUS
approved