# Thread: Complex Analysis - f is holomorphic...

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

Hey guys. Heres my problem:

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

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

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