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Math Help - Removable Singularity

  1. #1
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    Removable Singularity

    Having trouble with this question:

    Let f be analytic at a point w

    Define g(z) = \frac{f(z)+wf'(w)-zf'(w)-f(w)}{(z-w)^2}

    Show that g has a removable singularity at w.

    Please help!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Just show that the numerator has a zero of order at least 2 at w (can you see why this is what you have to do?).
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  3. #3
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    Um, I can't really see why that is. Also, not really sure how to go about it - I have to differentiate the numerator and show it's still zero?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Yes, exactly; show that the numerator is zero at w, and its derivative also.

    The idea is that these zeroes will cancel the zeroes of the denominator - if you only had a simple zero or no zero at all at w, you'd end up with a simple or a double pole there.
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  5. #5
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    Thanks, I think I've got it now
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