10.5.29 Conformal mappings

Problem: Find the image of domain \( D=\{\operatorname{Re} z<0\} \cup\left\{\frac{\pi}{2}<\operatorname{Im} z<\frac{3 \pi}{2}\right\} \backslash\{z=t+i \pi, t \in(-\infty ;-1]\} \) for the mapping \[ f(z)=\operatorname{sh} z \]