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Complex analysis question: Calculus of residues

  1. Dec 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f(z) = exp(2πiEz) / (1 + z^2), where E is some real number.

    Find the poles, their orders and the residues at each pole.


    2. Relevant equations



    3. The attempt at a solution
    Hi everyone, here's what I've done so far:

    1 + z^2 = (1 + i)(1 - i)

    Thus f has poles at z = i, z = -i, as f(z) is non-holomorphic at these points.

    Both are poles of order 1, as lim (z - a).f(z) are defined, for a = i, -i (limit as z goes to a)


    Now calculate the resdiues, use the formula:

    a_-1 = lim (z - a).f(z) [limit as z goes to a]


    and then Res f(z) = 2πi.a_-1


    ---

    Is this correct? Is it acceptable to just sub in the values into the equation like this, or are you expected to work out the Laurent series?

    Also, is there also a pole at z = infinity?
    Can it too be subbed into the equations above?

    Thanks for any help
     
  2. jcsd
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