I am working with the 2-dimensional form of the Coulomb potential $V_c=\dfrac{1}{\sqrt{x^2+y^2}}$ and in my calculation I eventually will need to compute the following Fourier Transform integrals:
\begin{equation} I_1 = \int e^{-i q y} \left(\frac{d}{dx} V_c\right) dy \end{equation} and \begin{equation} I_2 = \int e^{-i q y} \left(\frac{d}{dy} V_c\right) dy \end{equation}
I believe I can do the first one and its equal to $I_1 = \sqrt{\frac{2}{\pi }} \left| q\right| K_1\left(q \sqrt{x^2} \text{sgn}(q)\right)$ but am completely stumped on the second one. I cant seem to find it in any integral tables nor any remotely useful tricks I can use. Might anyone offer some ideas? Thanks in advance.