For the semi-disk with domain $$D=[x=(x_{1},x_{2}) \in\mathbb{R}^{2}: \lvert x \rvert^{2} < 1, \ x_{2} > 0 \}$$ how can one find the Green's Function for the upper half $ \ \ 0<r<1, \ 0<\theta<\pi$ using the method of images?
Can I use the general Green's Function for the full disk and subtract an image source at the bottom half of the disk? So probably,
$$\frac{1}{4\pi} \ln( \frac{r^2 + r_0^2 - 2rr_0cos\theta}{r^2r_0^2 + 1 - 2rr_0 cos\theta})-\frac{1}{4\pi} \ln( \frac{r^2 + r_0^2 - 2rr_0cos\theta_0}{r^2r_0^2 + 1 - 2rr_0 cos\theta_0})$$ if $x=(r.cos(\theta),r.sin(\theta))$ and $\xi=(r_0.cos(\theta_0),r_0.sin(\theta_0))$. For a Dirichlet problem, is this sufficient? Thanks in advance