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Math Help - Complex integration

  1. #1
    Junior Member enjam's Avatar
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    Complex integration

    Hey guys, quick question. How would I go about showing that this following equation:

    \int_{|z|=3}^{}tan(z).dz

    equals -4\Pi i

    ...I've tried breaking it down to sin(z) / cos(z), then finding the residues using the singularities that occur at \Pi /2 and -\Pi /2 but that just leads me to an answer of 0.

    If anyone could show me how to do this using the residue theorem, I'd really appreciate it. Thanks.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by enjam View Post
    Hey guys, quick question. How would I go about showing that this following equation:

    \int_{|z|=3}^{}tan(z).dz

    equals -4\Pi i

    ...I've tried breaking it down to sin(z) / cos(z), then finding the residues using the singularities that occur at \Pi /2 and -\Pi /2 but that just leads me to an answer of 0.
    If a function is of the form f(z)/g(z), and the denominator has a simple zero at z_0, then the residue there is given by f(z_0)/g'(z_0). In this case, f(z)/g'(z) = \sin z/(-\sin z) = -1. So the residue at both poles is 1, giving the integral as -4\pi i.
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