1
$\begingroup$

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?

Thanks.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.