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Thread: Laurent Series Question

  1. #1
    Mar 2012

    Laurent Series Question

    Ok, so this is just a question about clarification. I never learned this in class, so I was hoping someone could explain a small technicality.

    The question is Find the Luarent Series of 1/(1-z^2).

    I know how to do this. You can do it many ways, but I just first split it up into 1/(1-z)(1+z). Then I found that 1/(1+z) = 1/(2-(1-z)), and I just wrote out the series, divided by 1-z to get the Laurent sreies of
    1/(1-z^2) = -1/[2(z-1)]+1/4-1/8(z-1)+1/16(z-1)^2-...

    This should be right (but please correct me if I am wrong)

    However, the question then asks what happens when |z-1| > 2? I assume when |z-1| > 2, this series is no longer converging? Is that the case? Otherwise, why would they ask this?

    It seems to be when |z-1| > 2, the terms are like 1/4, 1/4, 1/4, and getting bigger?

    So I am confused. What am I suppose to write for the Laurent series then? Infinity?

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  2. #2
    Dec 2011

    Re: Laurent Series Question

    Your questions do not make any sense. The first question should be: Find the Laurent series for the function f about a point c.

    You have singularities at $\displaystyle z=1$ and $\displaystyle z=-1$, so you can do Laurent expansions in two different areas: $\displaystyle |z| < 1$ and $\displaystyle |z| > 1$ depending on where z is defined.

    I have no idea what you're asking in the second question since you can find a Laurent series expansion for f on $\displaystyle |z - 1| > 2$ that is convergent. If these are real question from a course, post them exactly as they were given to you.
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