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Holomorphic function convergent sequence

  1. Jan 22, 2017 #1
    1. The problem statement, all variables and given/known data

    Hi

    Theorem attached and proof.

    canidieyet.png

    I am stuck on

    1) Where we get ##|g(z)|\geq |a_m|/2 ## comes from
    so ##a_{m}## is the first non-zero fourier coeffient. So I think this term is ##< |a_m|r^{m}##, from ##r## the radius of the open set, but I don't know how to take care of the rest of the higher tems through ##a_{m}## , is this some theorem or?

    2) The conclusion thus ##f(z)## has only one zero at ##z=z_0##
    I think i'm being stupid but what is this being made from?
    We know ##g(z_0) = a_{m} \neq 0 ## and ##a_{0}=0##, but I dont understand.

    Thanks


    2. Relevant equations
    above
    3. The attempt at a solution
    above
     
  2. jcsd
  3. Jan 22, 2017 #2

    andrewkirk

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    The Lemma is very badly expressed. What do you think it means by 'if ##f(z_n)=0## for any ##n##, then...'? This vague phrase could either mean
    (1) ''if there exists some ##n\in\mathbb N## such that ##f(z_n)=0##, then...."
    or it could mean
    (2) "if for every natural number ##n##, ##f(z_n)=0##, then ...."

    The two interpretations have very different consequences.
     
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