# Indefinite integration of multiplication of two Bessel function

I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case?

$$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$

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The following may be helpful: For $a=0$ we have a known formula; writing $\int_0^\infty-\int_0^a$ we get a formula for your case; might be possible because $\int_0^a J_\mu(x)J_{\mu+1}(x)dx = \sum_{k \ge 0} J_{\mu+k+1}(a)^2$, though haven't given it more thought. –  Suvrit Jan 27 '14 at 16:57
It is not clear what you mean by "analytic expression". The integral that you wrote is an analytic expression (in my vocabulary). –  GH from MO Jan 27 '14 at 18:52
I mean analytic solution. Sorry for confusion. –  artalexan Jan 27 '14 at 19:03
It is not clear what you mean by "analytic solution". We are talking about an integral (not an equation), which is analytic (complex differentiable) in $a$. The word "solution" makes no sense in this context. –  GH from MO Jan 27 '14 at 19:11
OK, let me explain it in this way. Can you solve the integral? Find any F(x) in an explicit way, that you may put it on the right side. –  artalexan Jan 27 '14 at 19:35

For the case $b=c$ ... $$\int \!{{\rm J}_2\left(bx\right)}{{\rm J}_1\left(bx\right)}{dx}= \frac{1}{2b}-{\frac { \left( {{\rm J}_0\left(bx\right)} \right) ^{2}}{2b}}-{\frac { \left( {{\rm J}_1\left(bx\right)} \right) ^{2}}{b }}$$ (I used Maple.)
unfortunately in my case $b\neq c$ –  artalexan Jan 28 '14 at 8:45
Using formula (18.17) at this link here, you can get a power-series for $J_2(bx)J_1(cx)$, which you can simplify and integrate term-by-term to obtain a "closed-form" expression for your integral. Maple or Mathematica might be able to simplify that even further.