I am trying to solve the following integral (which can be seen also as a Hankel Transform):
$\int_{0}^{\infty} \frac{1}{\left( k-1 \right)^2} J_0(k r) \, dk$
where $J_0(k r)$ is the zeroth order Bessel function of the first kind, and $r$ is a positive real number.
My idea was to use integration by parts considering that $\frac{1}{\left( k-1 \right)^2} = \frac{d}{dk} \left(\frac{-1}{\left( k-1 \right)} \right)$ that is to say:
$\int_{0}^{\infty} \frac{1}{\left( k-1 \right)^2} J_0(k r) dk= \left[\frac{-J_0(k r)}{\left( k-1 \right)} \right]_{0}^{\infty}+r \, \int_{0}^{\infty} \frac{1}{\left( k-1 \right)} J_1(k r) dk$
however i cannot find any solution for the integral $\, \int_{0}^{\infty} \frac{1}{\left( k-1 \right)} J_1(k r) dk$
also, as this problem comes from a physical problem, it is important that the solution (which is a function of $r$) has to go to zero at $r \rightarrow \infty$ (however it can diverge at $r=0$), do you know if this can easily checked?
I checked Proudnikov Integrals and Series Vol.2 but I couldn't find anything helpful.
do you have any hint?
One can decompose $$I=\int_0^\infty \frac{J_1(kr)}{k-1}\,dk=-\int_0^\infty J_1(kr)\,dk+\int_0^\infty \frac{kJ_1(kr)}{k-1}\,dk$$. Which can be evaluated as$$I=-\frac{1}{r}-\int_0^\infty \frac{d}{dr} \frac{J_0(kr)}{k-1}\,dk$$ Though validity of the inversion of the order integration / derivation is unclear, using the expression you quote: $$\int_0^\infty \frac{J_0(kr)}{k-1}\,dk=\varphi(r) $$ you get$$I=-\frac{1}{r}-\varphi'(r)$$