10.5.35 Conformal mappings

Problem: Find the image of domain \[ D=\{\operatorname{Re} z<0\} \cup\left\{|\operatorname{Im} z|<\frac{1}{4}\right\} \backslash\{z=t,-1 \leq t<0\} \] for the mapping \( f(z)=\operatorname{sh} z \).