Complex Analysis: brach of the square root

  • #1
123
0

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.
 
Last edited:

Answers and Replies

  • #2
557
1
Use
Code:
"[itex]1+1=2[/itex]"
instead of
Code:
"[tex]1+1=2[/tex]"
 
  • #3
123
0
Thanks; I wondered what was up with that...
 

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