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I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various Schaum's) for this problem. I would like to do this by hand. Is it possible? I have had no luck with expanding out $J_n(x)$ and integrating term by term, as I cannot collect them into something nice at the end.

If possible and I just need to try harder (i.e. other methods or leaving it alone for a few days and coming back to it) that is useful information.

Thanks to anyone with a clue.

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    This is a good question, but I think it belongs on math.SE. I'm going to flag it for the moderators to move it there, but I'm interested in the answer myself. This would be good to know!2011-01-16

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