10.5.25 Conformal mappings

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