= Joukowski map from the upper half-disc to the upper half-plane
{c}
{title2=$J(w)=-(w+w^{-1})$}
The <holomorphic function> $J(w)=-(w+w^{-1})$ maps $D_+=\{w:|w|<1,\ \operatorname{Im}w>0\}$ conformally and bijectively onto the <upper half-plane>. Its imaginary part is $\operatorname{Im}w\,(|w|^{-2}-1)>0$, and its <derivative> $w^{-2}-1$ never vanishes there.
Equality of two values implies $(w_1-w_2)(1-1/(w_1w_2))=0$; since $|w_1w_2|<1$, this proves <injectivity>. For any target $\zeta$ in the <upper half-plane>, the quadratic equation $w^2+\zeta w+1=0$ has two <roots> with product one and neither on the <unit circle>. Exactly one therefore has modulus below one; the imaginary-part formula forces that <root> to lie in $D_+$, proving <surjectivity>.
In particular, $z\mapsto e^{iz}$ maps $\{0<\operatorname{Re}z<\pi,\ \operatorname{Im}z>0\}$ bijectively to $D_+$, and composition gives the <conformal map> $z\mapsto-2\cos z$ onto the <upper half-plane>. The two sides of the half-strip map to the outer real rays, while its bottom edge maps to the segment $[-2,2]$.
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