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    Contour Integrals: Maclaurin and Cauchy's Residue

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    Let C denote the circle |z| = 1, taken counterclockwise, and use the following steps to show that:

    int(exp(z + 1/z) dz) = i2pie sum(1/ (n!(n + 1)!)

    1) By using the Maclaurin series for e^z, write the above integral as

    sum(1/n! int(z^n exp(1/z) dz) )

    2) Apply Cauchy's residue theorem to evaluate the integrals above.

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    Solution Summary

    We compute a class of contour integrals using Cauchy's residue theorem as well as the MacLaurin series of the integrands.