10.5.44 Conformal mappings

Problem: Find the image of domain \( D \) when mapped by the Joukowsky transform: \[ D:\left\{\begin{array}{c} \operatorname{Re} z>0, \quad \operatorname{Im} z>0 \\ |z|>1 \end{array}, \quad f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)\right. \text {. } \]