10.5.34 Conformal mappings

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