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Homework Help: Laurent series(a really hard one!)

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a Laurent expansion valid for 0<|z-i|<1and use it to show that the derivative will leave you with the residue a₋₁.
    So we have an annulus with its center at one of the singularities. Since the other singularity at z=-i is outside of the annulus it is not a problem, because the function is analytic. .
    The problem is i have never found a laurent expansion for a function this complicated! with double poles!

    2. Relevant equations

    3. The attempt at a solution

    let w=z-i

    =((e^{iz})/4)*∑{n=0 to ∞}(-w)²ⁿ⁻²

    I cant figure this question out! so if anyone can help
  2. jcsd
  3. Oct 14, 2009 #2


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    Science Advisor

    Write the function as
    [tex]f(z)= \frac{e^{iz}}{z+i}\frac{1}{z- i}[/itex]
    As you say, [itex]e^{iz}/(z+ 1)[/itex] is analytic at z= i and so can be written as a Taylor series there. Multiplying that by 1/(z-i) only decreases each power of (z- i) in the series by 1.
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