I am trying to calculate the surface integral of a complex Log function i.e. $$ \int\int_{|z|<1}{Log(x+i y-(x_0+iy_0)))dxdy}$$ where $z=x+iy$ and $x_0,y_0 \in \mathbb{R}$ . I know that for analytic functions it is possible to calculate it like this $$ \int\int_{|z|<R} f(z)dxdy=\pi R^2f(0)$$ But I'm not sure how Log's singular point will affect the solution.
In addition, the Log function has a jump when completing a rotation which means it is not continuous, is this integral well defined?
Thanks in advance!