I have searched the literature on integrals over bessel functions, but I couldn't find anything. The integral to be evaluated is,

$\int_0^a J_{n}(bx)J_{\mu}(cx)xdx =: \mathcal{M}_r(a;n,\mu;b,c)$

where $J_n(x)$ is a Bessel function of integer order, $J_{\mu}(x)$ is a Bessel function of order $µ$, $a,b,c\in\mathbb{R}^+$. The RHS object denotes another expression having the arguments inscribed that solves the integral as good as possible.

A colleage couldn't get mathematica to yield a result, and I couldn't evaluate the integral using tables etc.

Can anyone please help?

Thanks in advance!


I have come up with following expression using books: 1) Integrals and Series - Special Functions, and 2) Table Of Integrals, Series And Products.

This looks different from what you exactly need, but with a simple transformation, you can find solution for when $b=c$. Further, if you deeply go through the literature, you can also find more general solution when $b\neq c$. Further, In your case $\lambda=1$: $$\int_0^a x^{\lambda } J_{\mu }(x) J_{\nu }(x) \, dx=\frac{a^{\lambda +\mu +\nu +1} }{2^{\mu +\nu } \Gamma (\mu +1) \Gamma (\nu +1) (\lambda +\mu +\nu +1)}\times\, _3F_4\left(\frac{1}{2} (\mu +\nu +1),\frac{1}{2} (\mu +\nu +2),\frac{1}{2} (\lambda +\mu +\nu +1);\mu +1,\nu +1,\mu +\nu +1,\frac{1}{2} (\lambda +\mu +\nu +3);-a^2\right)$$ which includes hypergeometric function.

  • $\begingroup$ Thanks a lot for your efforts! $\endgroup$ – David Heider May 9 '15 at 17:07

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.