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Existence of (complex) limit z->0 (z^a)

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Justify for which complex values of a does the principal value of [tex]z^a[/tex] have a limit as z tends to 0?


    2. Relevant equations

    [tex]z^a = e^{a log(z)} [/tex]

    [tex]log(z) = log|z| + (i) (arg(z)) [/tex]

    3. The attempt at a solution

    [tex]Lim_{z \rightarrow 0} z^a = Lim_{z \rightarrow 0} e^{(a) (log(z))} [/tex]

    [tex]=Lim_{|z| \rightarrow 0} e^{(a) (log|z|) + (i) (a) (arg(z))}[/tex]

    Let [tex]a = u + i v [/tex].

    [tex]=Lim_{|z| \rightarrow 0} e^{(u+iv) (log|z| + (i) (u+iv) (arg(z)))}[/tex]

    [tex]=Lim_{|z| \rightarrow 0} e^{(u) (log|z|) + (i) (v) (log|z|) + (i) (u) (arg(z)) - (v) (arg(z))}[/tex]

    [tex]=Lim_{|z| \rightarrow 0} e^{(u) (log|z|)} e^{(i) (v) (log|z|)} e^{(i) (u) (arg(z))} e^{-v (arg(z))}[/tex]

    [tex]=Lim_{|z| \rightarrow 0} |z|^u e^{(i) (v) (log|z|)} e^{((i) (u) - (v)) (arg(z))}[/tex]

    I just noticed a big mistake here, so I'm erasing this part. Any ideas?
     
    Last edited: Sep 22, 2009
  2. jcsd
  3. Sep 22, 2009 #2

    tiny-tim

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    Hi Knissp! :wink:

    Isn't it easier just to go polar, and put z = re ? :smile:
     
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