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Contour integral of a complex number

  1. Jun 3, 2014 #1
    Hello.
    I am stuck at the third point, that is from 1+i to i. I asked someone to show me his answer but that part of his is different from mine. Is his solution correct?
    Here it is:

    (i) z = 0 to 1 via z(t) = t with t in [0, 1]:
    ∫c1 Re(z^2) dz
    = ∫(t = 0 to 1) Re(t^2) * 1 dt
    = ∫(t = 0 to 1) t^2 dt
    = 1/3.

    (ii) z = 1 to 1+i via z(t) = 1+it with t in [0, 1]:
    ∫c2 Re(z^2) dz
    = ∫(t = 0 to 1) Re((1 + it)^2) * (i dt)
    = ∫(t = 0 to 1) (1 - t^2) * i dt
    = i(t - t^3/3) {for t = 0 to 1}
    = 2i/3.

    (iii) z = 1+i to i via z(t) = t+i with t in [0, 1] and opposite orientation:
    ∫c3 Re(z^2) dz
    = -∫(t = 0 to 1) Re((t+i)^2) * 1 dt
    = -∫(t = 0 to 1) (t^2 - 1) dt
    = -(t^3/3 - t) {for t = 0 to 1}
    = 2/3.

    (iv) z = i to 0 via z(t) = it with t in [0, 1] and opposite orientation:
    ∫c4 Re(z^2) dz
    = -∫(t = 0 to 1) Re((it)^2) * i dt
    = -∫(t = 0 to 1) -it^2 dt
    = i/3.

    So, ∫c Re(z^2) dz = 1/3 + 2i/3 + 2/3 + i/3 = 1 + i.
     

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  2. jcsd
  3. Jun 3, 2014 #2
    Aha, now I get it! It has an opposite orientation. I got it!
     
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