I am trying to figure out whether these integrals:
a) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y - \xi\,\right\vert\right)\,\right\vert\ \left\vert\,x - \xi\,\right\vert\ \left\vert\,y - \xi\,\right\vert}$$
b) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y - \xi\,\right\vert\right)\,\right\vert^{1/3}\ \left\vert\,x - \xi\,\right\vert\ \left\vert\,y - \xi\,\right\vert}$$
Converge, and if they do, show that the result does not depend upon $x,y \in \mathbb R^2$.
I am completely stumped. Would appreciate any ideas / hints in the right direction.
I would like to clarify that this is not a homework assignment, but rather an exercise to prepare for an exam.