# Thread: Analytic Function in a domain D

1. ## Analytic Function in a domain D

If f(z) is analytic in a domain D and maps D onto a portion of a straight line, then show that f(z) is a constant function in D.

Anyone know how to solve this?

2. ## Re: Analytic Function in a domain D

Do you know the open mapping theorem?

3. ## Re: Analytic Function in a domain D

Don't really because i just start to learn complex analysis.

4. ## Re: Analytic Function in a domain D

In fact the open mapping theorem was a sledgehammer proof. We can write $\displaystyle$f(z)=e^{i\theta z}a(z), where $\theta is a fixed real number. So$\displaystyle g(z):=e^{-i\theta}f(z)\$ is analytic on the open unit disk, and take only real values. By Cauchy-Riemann equations, what can you conclude?