Is "being harmonic conjugate" a symmetric relation?

Question

The question is:

Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, show whether or not the relation of being a harmonic conjugate is symmetric)

I'm pretty sure i'm correct in saying it isn't a symmetric relationship... But I'm wondering if someone can think of a direct counter-example to prove me right. Or is there an algebraic way to prove this that's better?

Any help is appreciated, thanks!

Answer 1

Here's the simplest example I can think of. $f(z)=z$ is clearly holomorphic, but $g(z)=Im(z)+iRe(z)$ is not, because when you look at the second Cauchy-Riemann equation you get $1 \neq -1$.

Answer 2

In general: $$f=u+iv\implies if = -v +iu,$$ i.e., $v$ is a harmonic conjugate of $u$ $\implies$ $u$ is a harmonic conjugate of $-v$.