10.5.21 Conformal mappings

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{c} \operatorname{Re} z>0 \\ 0<\operatorname{Im} z<\pi \end{array}, \quad f(z)=\operatorname{cth} z .\right. \]