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I am wondering how to do the integral $$\int_{0}^{k} u J_n^2({u}) du, \ n \in \mathbb{Z}^+,$$ and express my answer in terms of other first kind Bessel functions. I have searched here for useful identities, but I can't seem to combine them in a way that helps.

Integration by parts hasn't proved to be useful either, since from what I've tried, I have to evaluate either $\int u J_{n}(u) du$ or $\int J_{n}(u)^2 du$, which both can't be expressed in terms of Bessel functions alone.

Does anyone have any hints? Thanks :-)

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Hint:

Consider $$\dfrac{d}{dx}\left[x^2\left(J_{n}^2(x)-J_{n+1}(x)J_{n-1}(x)\right)\right]$$ and prove $$\int_{0}^{k} u J_n^2({u}) du=\dfrac12k^2\left(J_{n}^2(k)-J_{n+1}(k)J_{n-1}(k)\right)$$

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  • $\begingroup$ I was typing the same when came your answer. What is "surprizing" is that it seems that, only for $k=2$, $\int u J_n^k({u}) du$ shows a nice expression in terms of Bessel functions. Do you know any reason for that ? $\endgroup$ – Claude Leibovici Nov 5 '18 at 4:15

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