For figure $\mathbb{C} \setminus (\{ z \mid\mathrm{Im}(z) \leq 0\} \cup\{z = x+ iy \mid y \geq \sqrt{x^2 + 1}, x \leq 0\})$ find a conformal mapping to closed unit disk.
What I have done so far: I found a conformal map from this figure to $$\mathbb{C} \setminus (\{z \mid \mathrm{Im}z = 0, \mathrm{Re}z \leq 0 \} \cup \{z \mid \mathrm{Im}(z) \geq 0, \mathrm{Re}(z) \geq 1\}).$$ Then I tried to use Schwarz-Christoffel theorem but I'm stuck, because I have two vertices in infinity. However, I think I need the last step, and it can be done without Schwarz-Christoffel theorem...