I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:
$$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$
Where $n$, $\nu$, and $\mu$ are all integers, and $a$, $b$, and $c$ are all real and positive.
If not generally, a specific case would be quite helpful:
$$\int_0^\infty \mathrm{d}x~x^2 J_\nu(a x) J_\nu(b x) K_1(c x) $$
I am aware of the following:
- Gradshteyn & Ryzhik eq. (6.522.3), which calculates the following integral as a relatively simple function:
$$\int_0^\infty \mathrm{d}x~x K_0(ax) J_\nu(b x) J_\nu(c x)$$
- Gervois – integrals of three Bessel functions of the first kind: by manipulating the order of the last Bessel I could change the order of the last Bessel function, but I don't know how to "replace" it by $K_\mu$.
- Fabrikant – Computation of infinite integrals involving three Bessel functions
by introduction of new formalism: helpful formulas, but unfortunately his eq. (21) and (22) is not valid in the case above, so all the results in his discussion are useless to me as they start from eq. (21). He does use this formula to "replace" $K_\mu$ by $J_\mu$ in eq. (24), using a property relating the two functions (or at least something close to what this links to):
$$ \pi \mathrm{i} J_\nu(z) = \mathrm{e}^{-\nu\pi\mathrm{i}/2} K_\nu(-\mathrm{i} z) – \mathrm{e}^{\nu\pi\mathrm{i}/2} K_\nu(\mathrm{i} z), ~~~~|~\mathrm{arg}~z~| \leq \pi/2$$
I simply fail to see how this formula results in a simple change in his equations, especially because of the complex nature of the arguments introduced by the above formula, and the involved definition of $K_\mu$ for integer order.
Should I instead be looking for specific application of G&R eq. 6.522.17-18, which could provide the required formulas, or are there better approaches to this problem? It seems that Fabrikant in the above-linked article says the validity of these formulas is more strictly bounded than shown in G&R.
Any help here would be much appreciated.
Best Answer
Although I'm sure @Zurab's answer will indeed give me a solution (and I've heard the suggestion before in other situations), I'm not very familiar with the technique.
Following the suggestion by @Johannes in the comments, there is a much more straightforward way that, although not fully general, fits my problem exactly (i.e. it leads to expressions for the integrands I'm coming across).
Starting from G&R eq. (6.522.3), $$\int_0^\infty \mathrm{d} r~ r K_0(a r) J_\nu(b r) J_\nu(c r) = \left( \frac{r_2 - r_1}{r_2 + r_1} \right)^{|\nu|} \frac{1}{r_1 r_2}$$ where $r_1^2 = a^2 + (b-c)^2$ and $r_2^2 = a^2 + (b+c)^2$, one can use the derivative and symmetry properties of the bessel-$K$ functions to change the order from 0, incrementing the power of $r$ in the integrand at the same time. Simply calculating the derivative $c~\partial_c$ of the above righthand side gives my example from the question.