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Math Help - help for question on order of pole of function

  1. #1
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    help for question on order of pole of function

    Let D be a domain in \mathbb{C}, and let z_0 \in D. It is given that a function
    f : D\backslash \{z_0\}\rightarrow \mathbb{C} has a simple pole at z_0. Consider the function g given by

    g(z) = [f(z)]^2 for all z \in D \backslash \{z_0\}.

    Is it true that g has a double pole at z_0?



    This is my attempt.

    Since f has a pole of order 1 at z_0. Then
    f(z) = \frac{\phi(z)}{z-z_0} where \phi(z) is analytic at z_0 and \phi(z_0) \neq 0.

    Since g(z) = [f(z)]^2.
    g(z) = \frac{(\phi(z))^2}{(z-z_0)^2}.

    How do I proceed from here? Any idea or suggestion is welcome. Thanks in advance!
    Last edited by alphabeta89; October 26th 2012 at 02:41 AM.
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  2. #2
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    Re: help for question on order of pole of function

    \text{Define } \alpha(z) = \phi(z)^2, \text{ where } \alpha \text{ has the same domain } U \text{ as } \phi.

    \text{Then } g(z) = \frac{\alpha(z)}{(z-z_0)^2} \text{ on } U\backslash \{z_0\} \subset D\backslash \{z_0\}.

    \text{Is } \alpha \text{ holomorphic on the neighborhood } U \text{ of }z_0? \text{ Does } \alpha(z_0) = 0?

    \text{If you answered yes to both questions, then isn't that exactly the definition}

    \text{of } g \ (=f^2) \text{ having a pole of order 2 at }z_0?
    Last edited by johnsomeone; October 26th 2012 at 04:44 PM.
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  3. #3
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    Re: help for question on order of pole of function

    I'm sorry - I phrased that wrong.
    The point is that \alpha(z_0) \ne 0.
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