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I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral: \begin{equation} \int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\sin(a) J_\nu(a)-\cos(a) J_{\nu+1}(a) \right] \end{equation}

But I'm looking for something in the form $\int_0^1 x^\nu \sin(a x)J_\nu(b x)dx$, particularly for the case $\nu = 1$. I've been trying to get a solution but I haven't got any luck. Any ideas?

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  • $\begingroup$ Well, there doesn't appear to be a formula in DMLF so your last chance is Watson.. $\endgroup$
    – username
    Commented Oct 18, 2022 at 20:13
  • $\begingroup$ I tried it with Mathematica, it can do the first integral but gives up on your variant. $\endgroup$
    – Dan Romik
    Commented Oct 18, 2022 at 21:05
  • $\begingroup$ there is no closed form solution; you could expand in powers of $a$ and obtain a series expansion... $\endgroup$
    – Carlo Beenakker
    Commented Oct 19, 2022 at 11:03

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