F1[b_, r_] :=
1/(2 π)
Integrate[
x*BesselK[0, Sqrt[(r - x*Cos[y])^2 + x^2*Sin[y]^2]],
{x, 0, b}, {y, 0, 2 π}];
F2[b_, r_] := 1 - b*BesselK[1, b]*BesselI[0, r];
Using Mathematica, I can find that F1 and F2 give the same values when r <= b. But I cannot prove that F1 and F2 are identical. Any help?