Where does mapping $\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$ maps the unit disk $|z|<1$ in $w$- plane? given that $0 \leq \gamma < 2\pi$
Solution i tried - Given mapping is $$\;f(z)=\displaystyle \frac{1}{z} +e^{2i \gamma}z\;$$
we can write it as $$f(e^{i\theta})=e^{-i\theta}+e^{2i\gamma}e^{i\theta}$$
$$f(e^{i\theta})=\displaystyle e^{i\gamma}\left(e^{-i \theta}e^{- i\gamma}-e^{i \theta}e^{ \gamma}\right)$$
$$f(e^{i\theta})=\displaystyle2 e^{i\gamma}\cos(\theta+\gamma)$$
but after this i am not getting how to check its image in $w-$ plane ,because of that $\gamma$ , please help.
Thank you