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    $\int_\gamma z^n\,dz$ over circles centered at the origin without Cauchy's theorem

    P-TIE-S14-01

    Area
    Complex Analysis
    Status
    draft
    Contents

    The question provides some insight into Cauchy’s theorem. Solve the problem without using the Cauchy theorem.

    Evaluate the integral \(\int_\gamma z^n\,dz\) for all integers \(n\). Here \(\gamma\) is any circle centered at the origin with the positive (counterclockwise) orientation.

    This is part 1 of question 1 of the Spring 2014 list in the source. Part 2 (the same integral over circles not containing the origin) and part 3 (the integral of \(\frac{1}{(z-a)(z-b)}\) over \(\abs{z} = r\) with \(\abs{a} < r < \abs{b}\)) are separate cards.

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