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Homework Help: Series Question

  1. Apr 21, 2010 #1
    1. The problem statement, all variables and given/known data

    f(z) = (z + 2)/(z - 2)

    a) Find the Maclaurin Series for f on the doman |z| < 2.

    b) Find the Laurent Series for f centered at z0 = 0 on domain 2 < |Z| < inf.


    2. Relevant equations


    3. The attempt at a solution

    I'm having a hard time figuring out how (z + 2)/(z - 2) = 1 + (4/(z-2)) = 1 - (2/(1 - (z/2)).

    I tried referring to a geometric series, but I don't think I have the right approach.
     
    Last edited: Apr 21, 2010
  2. jcsd
  3. Apr 21, 2010 #2

    lanedance

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    hmmmm... been a while but how about
    [tex] \frac{z+2}{z-2} = (\frac{1/z}{1/z})\frac{z+2}{z-2} = \frac{1+2/z}{1-2/z} = (1+\frac{2}{z})\frac{1}{1-2/z} [/tex]
     
  4. Apr 21, 2010 #3

    lanedance

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    or for your specific question working back
    [tex] 1 + \frac{4}{z-2} = \frac{z-2}{z-2} + \frac{4}{z-2} = \frac{z-2+4}{z-2} = \frac{z+2}{z-2} [/tex]
    is that your question?
     
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