# Proving the identity of two functions

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?

• Since one of the functions is defined as a numerical evaluator, there is no way to make such a proof in Mathematica. I think you must reframe the question so that no numerics is involved. – m_goldberg Dec 24 '20 at 10:50
• I’m voting to close this question because the OP is asking for functionality that is not supported given the constraints the OP is putting on the solution. – m_goldberg Dec 24 '20 at 10:52
• You could try differentiating F2 to see if that gives the integrand of F1. – Sjoerd Smit Dec 24 '20 at 16:09