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Math Help - How to apply General Cauchy's Theorem

  1. #1
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    How to apply General Cauchy's Theorem

    The question was to evaluate the integral of f(z) dz, around C, where C is the unit circle centered at the origin, using the general cauchy's theorem.

    f(z) is 1/((2z^2)+1).

    I know that f(z) is not analytic at i(sqrt2)/2 and -i(sqrt2)/2 and both points happen to be inside C, so cauchy's theorem can't be applied.

    What's my next step?
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  2. #2
    Senior Member
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    Feb 2008
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    Re: How to apply General Cauchy's Theorem

    I don't see how you can use Cauchy's theorem here... unless maybe you mean Cauchy's residue theorem.

    But that's not necessary either. It's probably easiest to use Cauchy's integral formula. Draw a line through the unit circle such that each pole is inside one of the resulting half-circles. Let \gamma_1,\gamma_2 be appropriately-oriented paths corresponding to each half-circle such that \int_{|z|=1}f(z)dz=\int_{\gamma_1}f(z)dz+\int_{ \gamma_2}f(z)dz. Then by Cauchy's integral formulae we have

    \int_{|z|=1}f(z)dz=\int_{\gamma_1}\frac{1/2(z-[i/\sqrt{2}])}{z+[i/\sqrt{2}]}+\int_{\gamma_1}\frac{1/2(z+[i/\sqrt{2}])}{z-[i/\sqrt{2}]}

    =2\pi i\left[\frac{1}{2(-[i/\sqrt{2}]-[i/\sqrt{2}])}+\frac{1}{2([i/\sqrt{2}]+[i/\sqrt{2}])}\right]
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  3. #3
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    Re: How to apply General Cauchy's Theorem

    I haven't learned the Cauchy Residue Theorem yet.

    Thanks, I think I'm starting to get the hang of it now.
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