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Math Help - Residue Theorem

  1. #1
    Super Member Deadstar's Avatar
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    Residue Theorem

    Let f(z) = \frac{e^{i z}}{(1+z^2)^2}

    Find the singularities and classify them. Find the residue when z = i.

    Singularities found to be i and -i, both poles of order 2. So i figure ill need the taylor expansion at some point.
    Is there a set way to calculate residues? The question i did before this involved differentiating the denominator but that wont work here since there will still be a zero on the bottom line. So how to i find residues if they are of order 2?
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  2. #2
    Super Member Deadstar's Avatar
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    Actually i may have solved it, although its not at all like the working in the answer, though its does get the right one.

    Res(f, i) = lim_{z \rightarrow i} \frac{d}{dz} \frac{(z-i)^2(e^{iz})}{(z-i)^2(z+i)^2}. Then simplifying and differentiating the bottom line gives \lim_{z \rightarrow i} \frac{e^{iz}}{2(z + i)} = \frac{e^{-1}}{2i} = \frac{-i}{2e}.

    Which is the correct answer but not at all like the working given in the solutions...
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