10.5.10 Conformal mappings

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