# Hankel Transform integrals won't work in Mathematica

I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation:

$$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$

The answer is supposed to be

$$2\pi\mathrm i^m J_m(kr)$$

Mathematica cannot seem to do this integral; it simply gives me the input back:

Integrate[
Exp[I m phi] Exp[I k r Cos[phi]],
{phi, 0, 2 Pi},
Assumptions -> Element[m, Integers]
]

Why can't this be done? It can do it for explicit values of m = 0 and m = 1, but after that it begins reporting the answer in terms of polynomials times a zero-order Bessel function or something similar. Perhaps I'm expecting too much of the software, but I'd expect it to be able to verify integrals listed on Wikipedia pages. Is there something I'm overlooking?

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## migrated from stackoverflow.comFeb 18 '12 at 21:49

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I've had problems like this. If you only want to verify the known result, have you tried evaluating "Integral minus solution", to see if it simplifies to zero? – Jonas Heidelberg Feb 18 '12 at 13:23
I will wager that the reluctance of Mathematica to evaluate your integral generically is due to the differing behavior of the integral of the components Cos[m phi + k r Cos[phi]] and Sin[m phi + k r Cos[phi]], depending on whether m is odd or even. Try executing Assuming[k > 0 && r > 0, Table[Integrate[{Cos[m phi + k r Cos[phi]], Sin[m phi + k r Cos[phi]]}, {phi, 0, 2 Pi}], {m, 0, 5}]], for instance. – J. M. Feb 19 '12 at 0:08
"...after that it begins reporting the answer in terms of polynomials times a zero-order Bessel function or something similar." - yes, because it seems that Mathematica internally uses the recursion formulae for Bessel function within Integrate[] to simplify things. Unfortunately, it looks as if FullSimplify[] is ignorant of this very recursion formula, so... – J. M. Feb 19 '12 at 0:10