10.5.2 Conformal mappings

Problem: Find the image of domain \[ D=\{|z|>1\} \cup\left\{|z| \leq 1, I_{m} z>0\right\} \backslash\{z=i t,-4 \leq t<-1\} \] when mapped by the Joukowsky transform: \[ f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right) \text {. } \]