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    Evaluating an Integral With a 2nd Order Pole using the Residue Theorem

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    Evaluate the integral from 0 to INF of:

    (x^a)/(x^2 +4)^2 dx, -1 < a < 3

    We are to use f(z)= (z^a)/(z^2 +4)^2,
    with z^a = e^(a Log z), Log z= ln|z| + i Arg z, and
    -pi/2 < Arg z < 3pi/2.

    I have found the residue at 2i to be:
    [2^a(1-a)/16]*[cos ((pi*a)/2) + i sin ((pi*a)/2).

    Please let me know if this is correct and how to solve this problem.

    Many thanks for your help.

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    Solution Summary

    The residue theorem is used to evaluate an integral. The solution is well presented and includes a diagram.