F1[b_, r_] := 
  1/(2 π)      
      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?

  • 2
    $\begingroup$ 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. $\endgroup$ – m_goldberg Dec 24 '20 at 10:50
  • 1
    $\begingroup$ 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. $\endgroup$ – m_goldberg Dec 24 '20 at 10:52
  • $\begingroup$ You could try differentiating F2 to see if that gives the integrand of F1. $\endgroup$ – Sjoerd Smit Dec 24 '20 at 16:09

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