Right, so I'm struggling proving/disproving that for functions $u,v: \mathbb R^2 \to \mathbb R$ if $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (so the relation is symmetric)...
I've been trying to disprove it so far and I'm not having luck...
Am I correct in saying if a function is holomorphic, then it satifies Cauchy-Riemann equations and that is the best test for holomrophisms?
Also, if a function is indeed holomorphic, is that equivalent to saying it's analytic? Or is there a difference?
The definition of a harmonic function is as follows:
$u(x,y)$ is harmonic if there exists some $v(x,y) : f=u+iv$ is holomorphic, in which case, $v$ is the harmonic conjugate of $u$
If I'm trying to find a counter example to disprove my first statement, do I find 2 functions, $u , v$, substitute them into $f=u+iv$ and show that the Cauchy-Riemann equations don't hold? For some reason, I'm not believing that Cauchy-Riemann equations do test for holomorphism.
If you look at a similar question I posted, some user told me two different functions, and stated that one was holomorphic and one wasn't... but surely the fact that they're independently holomorphic is irrelevant... $f=u+iv$ has to be holomorphic right!?
And finally, if there are two holomorphic functions $u,v$, is it not the case that $f=u+iv$ will also always be holomorphic?
Any help is really really appreciated thanks