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Math Help - singularities and poles

  1. #1
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    singularities and poles

    http://www.math.ualberta.ca/~runde/files/ass411-6.pdf

    If I could get some help on #1 it would be much appreciated as I'm not too sure on how to start
    (unless you think the equation in the second part is a typo and it should be \frac{g(z)}{(z-z_0)^n})
    in which case I'll have another look since it would resemble my notes closer
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  2. #2
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    1\implies 2*. Since f is analytic on D it means in the neighorhood of z_0 we have that f(z) = \sum_{k\geq 0} \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k.
    However, f^{(k)}(z_0)=0 for 0\leq k < n. Thus, f(z) = \sum_{k\geq n} \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k = (z-z_0)^n \sum_{k\geq 0} \frac{f^{(n-k)}(z_0)}{(n-k)!}(z-z_0)^k.
    Therefore, f(z) = (z-z_0)^n g(z) where g(z) = \sum_{k\geq 0} \frac{f^{(n-k)}(z_0)}{(n-k)!}(z-z_0)^k and so g(z_0) \not = 0


    *)There appears to be a mistake it should say f^{(k)}(z_0) = 0 for k=0,1,2,...,n-1.
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