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 $-if(z) = v - iu$.