Complex Analysis: brach of the square root

[tarheelborn]

Homework Statement

Let [itex]f[/itex] be a quadratic polynomial function of [itex]z[/itex] with two different roots [itex]z_1[/itex] and [itex]z_2[/itex]. Given that a branch [itex]z[/itex] of the square root of [itex]f[/itex] exists in a domain [itex]D[/itex], demonstrate that neither [itex]z_1[/itex] nor [itex]z_2[/itex] can belong to [itex]D[/itex]. If [itex]f[/itex] had a double root, would the analogous statement be true?

Homework Equations

We say that a branch [itex]g(z)[/itex] of the [itex]p^{th}[/itex] root of [itex]z[/itex] exists on [itex]D[/itex] if [itex]g(z)[/itex] is continuous and [itex]g(z)^{p}=z[/itex] for every [itex]z \in D[/itex].

The Attempt at a Solution

I am really not sure at all how to begin this proof. I would appreciate a nudge to get started. Thank you.

 

[Quinzio]

Use

"[itex]1+1=2[/itex]"

instead of

"[tex]1+1=2[/tex]"

 

[tarheelborn]

Thanks; I wondered what was up with that...