Complex conjugate of $z$ without knowing $z=x+i y$

Solution 1:

As others have pointed out, since $\bar{z}$ is not holomorphic, there is no elementary function representation in terms of $z$.

The intuitive reason that complex conjugation is not holomorphic is that it reverses orientation.

More precisely, any holomorphic function is locally conformal (since the zeroes of its derivative are isolated). But conformal means, among other things, that orientation is preserved.

Since complex conjugation reverses the orientation of the entire complex plane, it can't be conformal, and thus it is not holomorphic.

Solution 2:

If you knew $z$ and $\bar{z}$ then you could easily recover the real and imaginary parts $x$ and $y$, since:

$x = \frac{z+\bar{z}}{2}$

$y = \frac{z-\bar{z}}{2i}$

So you certainly can't know $z$ and $\bar{z}$ without also knowing $x$ and $y$.