I must prove that a holomorphic function has a constant Re part if certain conditions are fulfilled.
We have that:
$f=u+iv$ is holomorphic in $D \subset C$, where C are all complex numbers. We have that $au+bv+c=0=>x=const.$ Prove it?
So I tried using Cauchy-Riemann: $$u_{x}=v_{y}$$ $$u_{y}=-v_{x}$$ but these statements don't lead me anywhere because we want to make use of the non-derivent value of these functions so I tried relating it to our equation: $$au+bv+c=0=>u_{x}=-\frac{b}{a} v_{x};u_{y}=-\frac{b}{a}v_{y}$$ and then when we relate the two things we have: $$u_{x}=v_{y}=-\frac{-b}{a}$$ $$u_{y}=v_{x}=\frac{b}{a}v_{y}$$ And that is my dead end, can someone help me out here?