10.5.14 Conformal mappings

Problem: Find the image of domain \[ D=\{-\pi<\operatorname{Im} z<3 \pi\} \backslash\{z=i t, \pi \leq t<3 \pi)\} \] for the mapping \( w=f(z) \), where \[ f(z)=e^{-\frac{z}{2}} . \]