Suppose $u$ and $v$ are real valued functions on $\mathbb{C}$. Show that if $v$ is a harmonic conjugate for $u$, then -$u$ is a harmonic conjugate for $v$.

I know I have to use cauchy reumann here. Not sure how to get started. Any hints or help will be greatly appreciate. Thanks in advance.

## Best Answer

The easiest way to see this is to realize that if $f(z) = u+iv$ is analytic, then so is $-if(z) = v - iu$.