10.5.28 Conformal mappings

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