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Math Help - Contour Integration

  1. #1
    Super Member Deadstar's Avatar
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    Contour Integration

    Here's a question a just did without any problems and the given solution for it.

    Evaluate \int_C |z|^2 dz if C is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) (traversed anti-clockwise, so that the vertices come in this order).

    (Solution is the attachment)

    Now my question is what would happen if it wasn't a square but a rectangle with vertices at (1,1), (2,1), (2,3), (1,3). How would i parametrize z?

    Here's what i thought it might be but I'm pretty sure its wrong, or at least some of them are. (You'll have to look at the given solution to see what AB, etc, represents)
    AB -> (i + t + 1) {perhaps should just be (i + t)}
    BC -> (2 + 2it)
    CD -> (3i + 2t)
    DA -> (1 + 2it)


    Any help on this please?
    Attached Thumbnails Attached Thumbnails Contour Integration-ci.jpg  
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Deadstar View Post
    Here's a question a just did without any problems and the given solution for it.

    Evaluate \int_C |z|^2 dz if C is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) (traversed anti-clockwise, so that the vertices come in this order).

    (Solution is the attachment)

    Now my question is what would happen if it wasn't a square but a rectangle with vertices at (1,1), (2,1), (2,3), (1,3). How would i parametrize z?

    Here's what i thought it might be but I'm pretty sure its wrong, or at least some of them are. (You'll have to look at the given solution to see what AB, etc, represents)
    AB -> (i + t + 1) {perhaps should just be (i + t)}
    BC -> (2 + 2it)
    CD -> (3i + 2t)
    DA -> (1 + 2it)


    Any help on this please?
    in general any line segment, say from P(a,b) to Q(c,d), is parametrized by x=ta+(1-t)c, \ y = tb + (1-t)d, \ 0 \leq t \leq 1, or briefly tP + (1-t)Q, \ 0 \leq t \leq 1,
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