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Math Help - Evaluate integral using Cauchy's Thm

  1. #1
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    Evaluate integral using Cauchy's Thm

    Evaluate:

    z
    ------- dz
    (z-2)^2

    and we're integrating from |z|=1. I'm asked to evaluate this using Cauchy's Thm, but I don't know how

    Can someone show the steps please. Thanks a lot
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  2. #2
    MHF Contributor chisigma's Avatar
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    The Cauchy's theorem extablishes that...

    \int_{\gamma} f(z)\cdot dz = 2\pi i \cdot \sum_{n} R_{n} (1)

    ... where R_{n} is the residue of each pole of f(*) inside \gamma. In Your case  f(z)= \frac{z}{(z-2)^{2}} and \gamma is the unit circle... how many poles of f(*) are inside \gamma?...

    Kind regards

    \chi \sigma
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  3. #3
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    I know what Cauchy's theorem says, but I don't know how to evaluate an integral using it.

    Can someone please post the steps showing how to do this?

    (I not here asking you guys to do my hw for me. I'm going to have a similiar question on my final and would like know how to go about this type of problem)

    Thanks in advance...
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  4. #4
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    Quote Originally Posted by jzellt View Post
    I know what Cauchy's theorem says, but I don't know how to evaluate an integral using it.

    Can someone please post the steps showing how to do this?

    (I not here asking you guys to do my hw for me. I'm going to have a similiar question on my final and would like know how to go about this type of problem)

    Thanks in advance...
    Do you know what a pole is? Do you know what a residue is? Do you know how to calculate a residue? These are things you need to know before attempting to use the Residue Theorem to integrate.

    Also, Cauchy's Integral Formula says:

    If f(z) is analytic in a region R containing a closed contour C and \alpha is any point enclosed by C then f^{(n)}(\alpha) = \frac{n!}{2 \pi i} \oint_C \frac{f(z)}{(z - \alpha)^{n+1}} \, dz.

    However, all this is irrelevant to your present question since the function you're integrating is analytic inside and on the closed contour |z| = 1 and so the value of the integral is zero.
    Last edited by mr fantastic; May 10th 2010 at 04:44 AM.
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  5. #5
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    Thanks for the input. Yes, I know what a pole and residue are...

    But I'm still not sure how to evalute this integral using Cauchy...

    Can someone show how? Thanks
    Last edited by jzellt; May 11th 2010 at 10:47 PM. Reason: Added a statement...
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