Regularized Green's function

Main Question or Discussion Point

Hello everyone,
I would like to know if there is a known, rigorous way to regularize a Green's function in coordinate space. In particular, it is known that the Green's function for a circle of radius R and source located at $\vec x_0$ is given by:
$$G(\vec x,\vec x_0)=\frac{1}{2\pi}\ln\left[\frac{\left|\vec x-\vec x_0\right|}{\left|\vec x-\frac{R^2}{|\vec x_0|^2}\vec x_0\right|}\frac{R}{|\vec x_0|} \right],$$
and therefore for $\vec x=\vec x_0$ diverges as $\frac{1}{2\pi}\ln\left|\vec x-\vec x_0\right|$. Is there any rigorous way of regularizing this function? The most natural way that is coming to my mind is clearly to subtract the divergence by simply defining:
$$G_R(\vec x,\vec x_0)=G(\vec x,\vec x_0)-\frac{1}{2\pi}\ln\left|\vec x-\vec x_0\right|.$$

Am I right? Is this rigorous?

Thanks a lot!

DuckAmuck