Let $\mathbb{H}\subset\mathbb{C}$ be the upper-half plane and let $$ G_0^\mathbb{H}(x,y) = \log\left|\frac{x-\overline{y}}{x-y}\right|,\text{ where }x,y\in\mathbb{H} $$
How do I show that
- $G_0^\mathbb{H}(x,\cdot)$ is harmonic is $\mathbb{H}\backslash\{x\}$, and
- As a distribution, $\Delta G_0^\mathbb{H}(x,\cdot) = -2\pi\delta_x(\cdot)$?
In other words, how do I show that $G_0^\mathbb{H}(x,\cdot)$ is the Green's function of the $\Delta_y$? I can't directly take the derivative of $G_0^\mathbb{H}(x,y)$ w.r.t $y$ since it involves complex conjugate so I am not sure how to proceed....
Not sure if it matters, but the boundary condition here is zero.