= Solution
The <Joukowski map> $f(z)=z+z^{-1}$ is <holomorphic> on $\mathbb C\setminus\{0\}$. A <holomorphic function> is locally <conformal> where its <derivative> is nonzero. Here $f'(z)=1-z^{-2}$ vanishes at $z=\pm1$. Therefore
$$
\boxed{\text{holomorphic on }\mathbb C\setminus\{0\};\qquad
\text{conformal at every point except }0,1,-1,}
$$
where the exclusion of zero is because the function is undefined there. Holomorphy alone does not supply conformality at the two <critical points>.
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