I have been doing some exercises as preparation for PDE exam and there was a task that required me to find a Green's function for $\Omega = D\left((1,0), \sqrt{2}\right) \cap D\left((-1,0), \sqrt{2}\right) \subset \mathbb{R}^2$, where $D\left((x,y), r\right)$ is a disc with center $(x,y)$ and radius $r$.
Now, I tried solving the task by finding a holomorphic map from $\Omega$ to unit disc, since for unit disc I know the Green's function and composition of Green's function with holomorphic one is again a Green's function, but since it has been quite a lot of time since I have been studying holomorphic functions, I struggle finding this particular one. The closest I got to is
$$\varphi(z) = \frac{z}{|z-i| |z+i| |z-(\sqrt{2}-1)| |z + (\sqrt{2}-1)|}.$$ The idea here was to send the ''boundary'' points, $(0,1)$, $(0,-1)$, $(\sqrt{2}-1,0)$, and $(-(\sqrt{2}-1),0)$ to infinity (plus or minus respectfully), but yet map $(0,0)$ to $(0,0)$. After mapping $\Omega$ to $\mathbb{R}^2$, I could map $\mathbb{R}^2$ to unit disc and since a composition of Green's function with holomorphic functions is Green's function, this would solve the problem. I am just so out of holomorphic functions that I have no clue whether this $\varphi$ is OK or not.
Any tips on how to find the holomorphic function from $\Omega$ to $\mathbb{R}^2$ would be appreciated, or if there is another, easier way to find the Green's function in this case, please let me know.