# Math Help - Complex Analysis - f is holomorphic...

1. ## Complex Analysis - f is holomorphic...

Hey guys. Heres my problem:

$f$ is holomorphic on $C$ and of the form $f(x + iy) = u(x) + i v(y)$ where $u$ and $v$ are real functions, then $f(x) = \lambda z + c$ with $\lambda \in R$ and $c \in C$.

Where $C$ is the coplex numbers and $R$ is the real numbers.

Earlier I've made an assignment where I show that if $f$ is holomorphic in a domain $G$ and $|f|$ is constant, then $f$ is constant. So I'm guessing I have to show that $f'$ is constant, which then tells me that there exists some $\lambda$ so $f(z) = \lambda z + c$, but I dont know how to do that.

If anyone could assist I would greatly appreciate it.

/Morten

2. ## Re: Complex Analysis - f is holomorphic...

Use the Cauchy-Riemann equations to show that u' and v' are constant.

3. ## Re: Complex Analysis - f is holomorphic...

Ah of course

Thanks for the help.

/Morten