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    Question: Find the laurent series expansion about z=0 for the function

    f(z) = (z+1)/(z(z-2)) 0<|z|<2

    What is the principle part of the series?

    I believe the answer is -1/2z - 3/4 -3z/8 -3z^2/16 with principle part -1/2z.

    Is this correct?
    How does the interval 0<|z|<2 effect the series?


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    It's ok as long as you weren't expected to find the general term.

    The series will diverge for |z| > 2, and has an obvious singularity at z = 0.
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    What do you mean by the general term?
    Is there also a singularity at z=2?
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    (Original post by 123student456)
    What do you mean by the general term?
    Is there also a singularity at z=2?
    Well, 1/(1-z) has series 1 + z + z^2 + ..., so the "nth term" is z^n.

    Or to put it another way, I can write:

    \displaystyle \frac{1}{1-z} = \sum_0^\infty z^n with equality when |z| < 1.

    In contrast, your series doesn't actual equal \dfrac{z+1}{z(z-2)} for any z other than 0 or 2.
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    (Original post by 123student456)
    What do you mean by the general term?
    Is there also a singularity at z=2?
    The series is actually divergent for all z with |z| = 2. I'm not sure you can meaningfully say a series has a singularity where it's non-convergent. Don't forget you asked about the series, not the function - they don't actually behave the same on |z|=2.
 
 
 
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Updated: February 12, 2017
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