Is there any way to attain the fundamental solution of 2D Poisson's equation $\nabla^2u(\mathbf{x})=\delta(\mathbf{x})$ using the theorem of complex functions?
I think a candidate solution could be to assume a complex function $f(z)=u(z)+1i\times v(z)$. based on the Cauchy-Riemann condition $\nabla^2u(z)=0$. Therefore, in the $\nabla^2u=\delta(z)$, the point $z=0$ is a non-analytic point.
However, I have no idea on how the functional form of the fundamental solution $u(\mathbf{x})=\frac{1}{2\pi}\ln|\mathbf{x}|$ could be derived from complex-function analysis.