10.5.20 Conformal mappings

Problem: Find the image of domain \( D=\{|\operatorname{Re} z|<\pi\} \cup\{\operatorname{Im} z>0\} \backslash\{z=i t, t \in[1 ;+\infty)\} \) for the mapping \[ f(z)=\sin \frac{z}{2} \text {. } \]