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Complex Analysis: brach of the square root

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data
    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?


    2. Relevant 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].


    3. 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: Nov 29, 2011
  2. jcsd
  3. Nov 29, 2011 #2
    Use
    Code (Text):
    "[itex]1+1=2[/itex]"
    instead of
    Code (Text):
    "[tex]1+1=2[/tex]"
     
  4. Nov 29, 2011 #3
    Thanks; I wondered what was up with that...
     
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