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    Complex Integrals - Cauchy's Integration Formula

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    Three questions to do with Cauchy's integral formula are evaluated in clear, easy to understand steps.

    1. Show that
    see attached
    where C is the semicircule |x|=R>1, lm(x)>=0 from x=-R to z=R.

    2. Use Cauchy's integral formula for derivatives to evaluate
    see attached
    where |z-2|=2 is oriented clockwise.

    3. Evaluate
    see attached
    where C is the square with vertices 2+/-2i, -2+/-2i, oriented counterclockwise

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    Solution Summary

    This solution reviews some basic properties of line integration, including the Cauchy integration formula for derivatives. Then a detailed solution is presented for each of the three questions to show by example how to apply these properties.