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Math Help - Complex Contour Integral

  1. #1
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    Complex Contour Integral

    I'm attempting to do a complex contour integral as follows:

    Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

    I parameterized:

    z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

    Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)

    I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
    =integral from 0 to 2pi i+i/9e^2iƟ dƟ

    iƟ + -1/18*i*e^-2iƟ

    which when evaluated from 0 to 2pi gives me

    [2pi*i - 1/18 * i * 1] - [0- 1/18*i]

    = 2 * i * pi for my final answer.

    (since e^-2pi*i = e^0= 1)


    My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Negativ View Post
    I'm attempting to do a complex contour integral as follows:

    Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

    I parameterized:

    z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

    Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need to show this is piecewise continuous or is it already taken care of?)
    From a to b? Not really sure what a to b is, but I assume you mean \int_{0}^{2\pi}f(e^{i\theta})z'(\theta )d\theta. You can think of z as a function on theta, or you can just think of it as a change of variables. In other words, if z = 3e^{i\theta}, dz = 3ie^{i\theta}d\theta.

    Quote Originally Posted by Negativ View Post

    I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ
    =integral from 0 to 2pi i+i/9e^2iƟ dƟ

    iƟ + -1/18*i*e^-2iƟ

    which when evaluated from 0 to 2pi gives me

    [2pi*i - 1/18 * i * 1] - [0- 1/18*i]

    = 2 * i * pi for my final answer.

    (since e^-2pi*i = e^0= 1)
    Okay. I believe you. I couldn't read any of that because of the lack of latex input.

    Quote Originally Posted by Negativ View Post

    My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

    Thanks in advance.
    Again, really not sure what you wrote above. But, I can assure you that

    \int_{c} \big[\frac{1}{z} + \frac{1}{z^3}\big]dz = 2i\pi.
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  3. #3
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    Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

    Edit: How would one type in the more easily readable text?
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  4. #4
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    Quote Originally Posted by Negativ View Post
    Thank you very much for the response. What you interpreted from my spastic text was what I was saying.

    Edit: How would one type in the more easily readable text?
    Write in latex code, and wrap the code in [ MATH ] and [ /MATH ] tags.

    LaTeX:Symbols - AoPSWiki
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