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Math Help - residues 2

  1. #1
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    Smile residues 2

    Ok, so I have this function and I need to find the residues for it.
    I developed the function into Laurent series and my question is, how can I find a(-1) which is the residues?

    Thanks in advance.
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  2. #2
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    There is only one singularity, at z = 2. It is an essential singularity, but there are only even powers of z2 in the Laurent series. So there is no term in (z-2)^{-1} and therefore the residue is 0.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    There is only one singularity, at z = 2. It is an essential singularity, but there are only even powers of z2 in the Laurent series. So there is no term in (z-2)^{-1} and therefore the residue is 0.
    Wait a second, if it's an essential singularity, does that means that the residue is 0 or does not exist?

    How about this one?

    Thanks again.
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  4. #4
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    I, myself, would have said that if a point is an essential singularity the function does not HAVE a residue there but I note that Wikipedia asserts that the residue at an essential singularity, z_0 is still the coefficient of (z- z_0)^{-1} in the Laurent series only noting that in this case, that is the only way to find the residue, the other formulas, such differentiating n times where n is the order of the pole, not applying. I guess I've been wrong all these years!

    Therefore, as Opalg says, the Laurent series for your original function having only even powers, the coefficient of (z-2)^{-1} is 0 and the residue is 0.

    For your second function, the coefficient of z^{-1} is 1, as you show, and so the residue is 1.
    Last edited by mr fantastic; February 10th 2009 at 06:08 AM. Reason: Fixed some latex
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  5. #5
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    Quote Originally Posted by asi123 View Post
    Wait a second, if it's an essential singularity, does that means that the residue is 0 or does not exist?
    If you define the residue of f(z) at z=a to be the coefficient of (z-a)^{-1} in the Laurent series about that point, then what I said before is correct. However, I see that Ahlfors defines it as "the unique complex number R which makes f(z) - R/(z-a) the derivative of a single-valued function in an annulus 0<|z-a|<\delta." Ahlfors is evasive about whether such a number exists in the case of an essential singularity, commenting that in this case "there is no [...] procedure of any practical value" for computing residues, "and thus it is not surprising that the residue theorem is comparatively seldom used in the presence of essential singularities."
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