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
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 06 2016
STATUS
approved