10.5.14 Conformal mappings

Problem: Find the image of domain $$D=\{-\pi <\mathrm{Im}z<3\pi \}\mathrm{\setminus }\{z=it,\pi \le t<3\pi )\}$$ for the mapping $w=f(z)$, where $$f(z)=e^{-\frac{z}{2}}.$$