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Complex Integration

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Sketch the C1 paths a: [0; 1] -> C, t -> t + it2 and b: [0; 1 + i]. Then compute the following integrals.

    ∫Re(z)dz over a

    ∫Re(z)dz over b


    2. Relevant equations



    3. The attempt at a solution

    Sketching a seems ok, y axis is Imaginary, x axis is Real, and the path is a quadratic between 0 and 1.

    However I'm not sure about b...

    As for the integrals, is it just a case of integrating Re(a(t))a'(t)dt between 0 and 1, then integrating Re(a(t))a'(t)dt between 0 and 1+i?
     
  2. jcsd
  3. Nov 26, 2011 #2

    I like Serena

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    Hi MM!

    I believe you've got (a) down.

    For (b) I have to admit I haven't seen the notation [0; 1 + i] before.
    Do you have a definition for it?
    I'd assume it's supposed to represent the line segment between 0 and (1+i).
    Or in other words: (1+i)[0, 1].
    It looks a bit weird though, since a curve is usually defined on an interval of real numbers.
    However, if this is the case, you need to reconsider what Re(z) is.
     
  4. Nov 27, 2011 #3
    I don't have a definition for it. I'll upload the actual problem to show it to you.
     

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  5. Nov 27, 2011 #4

    I like Serena

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    I get it, in particular if I look at the problems that are coming, that show which theory you're currently learning.

    Path b is not related to path a.
    It is just a (linear) path from 0 to (1+i).
    It would be given by b:[0,1]→C defined by t→t(1+i)

    Note that path a also starts in 0 and ends in (1+i).

    And also note that for path a the interval is [0,1] and not [0;1].
    [0,1] is a real interval, whereas [0;1] would indicate a (linear) path between 0 and 1 in the complex plane.
     
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