Find Green's function for $\Omega = \{(x,y) \in \mathbb{R}^2\mid x^2+y^2 < r^2, y>0 \} $.
I tried to find it and I know that $G(x,y)=\frac{1}{2}\pi\cdot \ln\left(\frac{1}{|y-x|}\right) + h^x(y)$. But I don't know how to find the $h^x(y)$ function. Could someone help please?
This is usually done by reflection (aka the method of images). Reflect the half-disk across the diameter, extending it to the disk $D$. For the disk, Green's function is known (and easy to obtain from a Möbuis map): $$G_D(x,y)=\frac{1}{\pi}\ln\frac{|1-\bar y x|}{|y-x|}$$ (It's somewhat unusual to use $x,y$ for complex numbers, by the way.)
The idea is to subtract $G(x,\bar y)$, because $\bar y$ is the reflection of $y$ in the diameter. The symmetry makes sure that the difference vanishes on the diameter, and so $$G_D(x,y)-G_D(x,\bar y)$$ is Green's function for half-disk $\Omega$. (The extra pole at $\bar y$ is outside of $\Omega$, so it does not concern us.)