Is there a closed form solution for the following integral

$$ \int_0^a dx \ x \ J_1(x) \ , $$ where $J_1(x)$ is a Bessel Function and $a>0$ is a finite real number?


  • $\begingroup$ Ask Wolfram Alpha. $\endgroup$ – Yves Daoust Apr 24 '16 at 20:13
  • $\begingroup$ Thanks, @YvesDaoust, I don't know why I didn't check that myself in the first place. I am not familiar with Struve Functions, so if anyone has a different way of getting to the solution, it would be appreciated. $\endgroup$ – Julien V Apr 25 '16 at 13:26

The solution, according the Mathematica, is

$$ \int_0^a dx \ x \ J_1(x) = \frac{\pi a}{2} \left( H_0(a)J_1(a) - H_1(a)J_0(a) \right) \ . $$

$H_n(x)$ is the Struve Function.

  • $\begingroup$ For even $n$, you can get the integral $\int x J_n(x)\;dx$ not using Struve. $\endgroup$ – GEdgar Apr 25 '16 at 13:38

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