I am trying to evaluate the following integral in Mathematica:

Integrate[Sin[θ] Exp[-I k (z - zp) Cos[θ]] 2 π BesselJ[0, k Sqrt[(x - xp)^2 + (y - yp)^2] Sin[θ]], {θ, 0, π}],

which just generates the input. I have tried applying a FullSimplify to the argument before the evaluation but it does not simpplify further.

Is there any methods that I could use to produce an evaluation of this? The problem seems to be originating due to the presence of the Euler term Exp[-I k (z - zp) Cos[θ]], without which the integral does evaluate.

  • 2
    $\begingroup$ Not all integrals can be expressed with the usual functions we work with. That's why people introduce special functions (such as the Bessel ones): the diff eq BesselJ comes from doesn't have a solution expressible in terms of elementary functions. There may be no practically useful symbolic solution to your integral, but that doesn't mean that you can't compute it numerically (with Mathematica) or that you can't prove various properties of the result (by hand, not done automatically by Mma). $\endgroup$ – Szabolcs Jan 8 '16 at 12:20

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