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Math Help - Isolated Singularities Foundation Questions

  1. #1
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    Isolated Singularities Foundation Questions

    In my class we are learning about isolated singularities, here is my understanding:

    1. Removable Singularity: a function f that is analytic in an open disc except for at the origin.
    Property:
    a) f is bounded in the neighorhood of that point.
    b) All positive terms.

    2. Pole of Order m: the coefficient of the -1 term of the power series.
    Property:
    a) |f(z)| \rightarrow \infty
    b) Finite many negative terms.

    3. Essential Singularity: A function f with Laurent series that has infinitely many negative terms.
    Property:
    a) The image f(U) is dense in complex plane.

    Are my properties listed correctly?

    Here are my questions:

    Why is property 2a true?

    If a function f is completely holomorphic in a disc D, including its center, then it doesn't have a singularity, right?

    This question is more in general, but how does a meromorphic function tie to all this? A meromorphic can only be a pole?

    Thank you.
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  2. #2
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    A meromorphic function is analytic everywhere except at poles. If a function has a pole of order  m , then we can write it as:  f(z) = \frac{g(z)}{(z-z_0)^{m}} . Now as  z \to \infty ,  |f(z)| \to \infty .
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