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Complex Variable Analysis

  1. Oct 21, 2007 #1
    I'm supposed to evaluate where a function is a regular point, essential singularity, or a pole (and of what order) at a specific location.

    Problem here.

    Evaluated at (where else but) z=1.

    I know its not a regular point since it doesn't evaluate to a simple Taylor Series... likewise -- I know its not an essential singularity since its Lorentz series doesn't go on forever (unless I'm entirely wrong).

    I cannot tell what order the pole is though:

    Using l'Hopital's rule (0/0) would suggest that its a pole of order 1... but applying it directly which the textbooks seem to do would suggest its a pole of order 2.

    It would be easier if I could conform it to a Lorentz series but across 7 textbooks I have on the topic of Complex Analysis (not kidding -- I just bought one today -- 120$ -- only a dozen problems on the topic -- and no solutions nor even answers). "Arfken" -- good for reference -- but I really should have bought Boas.
     
  2. jcsd
  3. Oct 21, 2007 #2
    A pole is simple if the limit

    [tex]\lim_{z\rightarrow z_0}(z-z_0)f(z)[/tex]

    exist and is finite.

    A pole is of order [itex]n[/itex] if the limit

    [tex]\lim_{z\rightarrow z_0}(z-z_0)^n f(z)[/tex]

    exist and is finite.

    (Why?)

    ---EDIT---

    You should read Marsden's or Churchill's book on the topic. Another good, but more advanced reference is Ahlfors.
     
    Last edited: Oct 21, 2007
  4. Oct 21, 2007 #3
    I would think that your pole is of order one, since your function is equal to (z+1)/(z-1)
     
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