Is the conjugate of $z$ a conformal map?

Question

Let $f(z) = \overline{z} $. At which points is $f(z)$ conformal?

I believe it is not conformal since $f$ is not analytic: It does not satisfy the Cauchy-Riemann equations. Is this correct?

Answer 1

Depends on the definition of conformal map.

Some authors define conformality as preserving angles both in size and orientation. Then $f(z)=\bar z$ is not conformal.

Others require only preservation of angles, and then $f(z)=\bar z$, as other anti-holomorphic functions, is included among conformal maps.

The second point of view may be less common in complex analysis textbooks, but it makes sense if one also thinks of the higher dimensions, where Möbius transformations are the only conformal maps we have. The inversion $x\mapsto x/\|x\|^2$ is orientation-preserving in $\mathbb{R}^n$ when $n$ is odd but is orientation-reversing when $n$ is even. It would seem odd to call it conformal only half of the time. Besides, the angles between lines can't be given a sign/orientation in $\mathbb{R}^n$ when $n>2$.