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?

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- Mar 27th 2012, 07:12 AMyounhockAnalytic 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? - Mar 28th 2012, 07:47 AMgirdavRe: Analytic Function in a domain D
Do you know the open mapping theorem?

- Mar 28th 2012, 07:49 AMyounhockRe: Analytic Function in a domain D
Don't really because i just start to learn complex analysis.

- Mar 28th 2012, 07:54 AMgirdavRe: 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?