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Math Help - Basic Complex Analysis question

  1. #1
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    Basic Complex Analysis question

    This question appears really easy, I am just not sure I am undestanding it correctly.

    I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
    Last edited by thecow; November 29th 2007 at 04:16 PM.
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  2. #2
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    Quote Originally Posted by thecow View Post
    This question appears really easy, I am just not sure I am undestanding it correctly.

    I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
    You need the version of the residue theorem with winding number.

    RonL
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    We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)
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    Quote Originally Posted by CaptainBlank View Post
    You need the version of the residue theorem with winding number.

    RonL
    Quote Originally Posted by thecow View Post
    We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)

    You can do what CaptainBlank said and use the residue theorem. But here is a special case, you can rely on the Cauchy integral formula but you still need winding numbers because \Gamma is not a contour (i.e. a simple closed curve).
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  5. #5
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    Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?
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  6. #6
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    Quote Originally Posted by thecow View Post
    Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?
    Yes. Just count the number of times it "winds" around the number.
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  7. #7
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    Alright thanks. I thought it might be 2 because if you follow the path with your finger you really only go around twice it seems.
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