I'm interested in solving this problem in closed form, if such a solution exists, in $2d$.
$$\int_{\Gamma_R} \frac{\partial G}{\partial \vec{n}} d\Gamma_R$$
Where $\Gamma_R$ is a circle of radius $R$ centered at the origin. $\vec{n}$ is the exterior normals of $\Gamma_R$.
Let $G$ be the fundamental solution of the laplace equation ($\Delta u = 0$), in $2d$.
$G(x, \bar{x}) = -\frac{1}{2 \pi} ln\left(\lVert x - \bar{x}\rVert\right)$
$\frac{\partial G}{\partial \vec{n}} = \nabla G \cdot \vec{n}$
$\nabla G = -\frac{1}{2 \pi} \frac{x - \bar{x}}{\lVert x - \bar{x} \rVert^2}$
Given that our domain is a circle, the exterior normals should be $\vec{n(x)} = \frac{x}{\lVert x \rVert}$
Adding it up together:
$$\int_{\Gamma_R} \frac{\partial G}{\partial \vec{n}} d\Gamma_R = -\frac{1}{2 \pi} \int_{\Gamma_R} \frac{(x - \bar{x}) \cdot \vec{n}}{\lVert x - \bar{x} \rVert^2} d\Gamma_R = -\frac{1}{2 \pi} \int_{\Gamma_R} \frac{(x - \bar{x}) \cdot x}{\lVert x - \bar{x} \rVert^2 \lVert x \rVert} d\Gamma_R$$
I also looked at transforming this into polar coordinates which would faciliate the integration over a circle, however I'm not quite sure how to proceed.
Also the point $\bar{x}$ should be inside the circle, if that makes any difference.