Find a conformal map from $F=\{z=x+iy\in\mathbb{C}:-\pi<y<\pi\}\setminus(-\infty,0]$ to the upper half plane $\mathbb{H}=\{z=x+iy\in\mathbb{C}:y>0\}$
To be honest, I never dealt with strips with slits before so I don't really have an idea where to start. The first idea that came to mind is maybe use the logarithm function (since its' principal branch is when we take $\mathbb{C}\setminus(-\infty,0]$. Other than that I don't know how to continue. What is the general way to think on such questions?
It helps to draw a picture of the image of each conformal map you use until you "reach" the upper half plane. Then the composition of those maps will be a conformal map onto $\mathbb{H}$.
First note $\exp$ maps $F$ onto the slit plane $\mathbb{C}\setminus(-\infty,1]$. The transformation $z\mapsto -z$ maps $\mathbb{C}\setminus (-\infty, 1]$ onto $\mathbb{C}\setminus [-1,\infty)$. Translation by $+1$ maps $\mathbb{C}\setminus[-1,\infty)$ onto $\mathbb{C}\setminus[0,\infty)$. Finally, a square root map sends $\mathbb{C}\setminus [0,\infty)$ onto $\mathbb{H}$.