Is there a way to (at least partially) integrate a combination of Bessel functions like:
$$\int_0^{\infty}J_m(b x)- \frac{J_m(a x)}{H_m^{(1)}(a x)}H_m^{(1)}(b x)\mathrm{d}x$$
Since $H_m^{(1)}(x)H_m^{(2)}(x)=J_m^2(x)+Y_m^2(x)$, I tried to use identities like: [(19) in paragraph 19.3 of table of integral transforms by Bateman (see http://authors.library.caltech.edu/43489/)], which reads:
$$\int_0^\infty \frac{J_0(a x)Y_0(b x)-J_0(b x)Y_0(a x)}{J_0^2(b x)+Y_0^2(b x)}\frac{x}{\lambda^2+x^2}\mathrm{d}x=-\frac{\pi}{2}\frac{H_\nu^{(2)}(a\lambda)}{H_\nu^{(2)}(b\lambda)}$$
Unfortunately this particular one seems to be wrong and I wasn't able to spot the error. Has anyone out there seen similar relations or integral representation of reciprocal Bessel function (of third kind) or knows an erratum for the equation above?
Any help is appreciated!
Thanks a lot