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Poles, zeroes, and essential singularities of z^(2/3)

  1. Sep 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Identify the zeroes, poles and essential singularities of [tex]f(z) = z^{2/3}[/tex]

    2. Relevant equations

    [tex]f(z) = e^{\frac{2}{3}\log{z}}[/tex]

    Which I'm not sure will be useful....

    3. The attempt at a solution

    I know that f is 0 at [tex]z=0[/tex], but what is the order of this zero? Is there such a thing as a fractional order zero? Conversely, I know that there is a singularity at [tex]z=\infty[/tex], but this singularity seems to be a fractional order pole. Is this the case? I know that both of these points are also branch points, does that influence the problem at all?

    Thanks for your help!
  2. jcsd
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