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    Complex variables: Contour interval, Taylor series

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    ** Please see the attachment for the full equation **

    Find the contour integral (please see the attached file)
    where [gamma] is the circle |z| = 4 oriented counterclockwise.

    Find the Taylor series of the function f at z_0 and the derivative f^(50) (z_0) when:
    1) f(z) = (2z+1)/(z(z-2) at z_0 = 1
    2) f(z) = sin z at z_0 = 0
    3) f(z) = e^z^2 at z_0 = 0

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    Solution Preview

    ** Please see the attached file for the complete solution **

    1. We wish to evaluate the contour integral:
    (please see the attached file)

    where (please see the attached file) is the contour (please see the attached file) going counterclockwise.
    (please see the attached file)

    Since (please see the attached file) is entire, the only poles of the integrand are 0 and (please see the attached ...

    Solution Summary

    In this solution we solve problems involving contour integration and Taylor expansion.