# Analytic Function in a domain D

• March 27th 2012, 07:12 AM
younhock
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?
• March 28th 2012, 07:47 AM
girdav
Re: Analytic Function in a domain D
Do you know the open mapping theorem?
• March 28th 2012, 07:49 AM
younhock
Re: Analytic Function in a domain D
Don't really because i just start to learn complex analysis.
• March 28th 2012, 07:54 AM
girdav
Re: Analytic Function in a domain D
In fact the open mapping theorem was a sledgehammer proof. We can write $f(z)=e^{i\theta z}a(z)$, where \$\theta is a fixed real number. So $g(z):=e^{-i\theta}f(z)$ is analytic on the open unit disk, and take only real values. By Cauchy-Riemann equations, what can you conclude?