10.5.9 Conformal mappings

Problem: Find the image of domain \( D \) for the mapping \( w=f(z) \), where \[ D:\left\{\begin{array}{l} 0<\operatorname{Re} z<\pi, \quad \operatorname{Im} z<0 \\ z \neq \frac{\pi}{2}+i t, \quad t \in[-1 ; 0) \end{array}, \quad f(z)=e^{i z} .\right. \]