10.5.24 Conformal mappings

Problem: Let's find the image of domain \( D=\{\operatorname{Re} z<0\} \cup\{|\operatorname{Im} z|<\pi\} \backslash\{z=t, t \in[-1 ; 0)\} \) for the mapping \[ f(z)=e^{\frac{z}{2}} . \]