Image of Upper Unit Semi Circle under Joukowsky Transformation

Question

I'm trying to understand how the Joukowski Transformation would map the following region:

$$\{z|0<arg(z)<\pi , |z|<1\}$$

with the Joukowski Transformation being : $w = \frac{1}{2}(z+\frac{1}{z})$

I know that the transformation maps the boundary of the unit circle on the $[-1,1]$ segment on the real axis of the $w$-plane but I don't know how to find the rest.

Any thought?

Many thanks !

Answer 1

Hint:

Parametrize $$x=\cos\theta,$$ $$y=\sin\theta,$$ where $0\le\theta\le2\pi$ for the complex number $z=x+iy$.

Then try to separate the Re and Im parts of $\dfrac{1}{2}\left(x+iy+\dfrac{1}{x+iy}\right)$.