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Parametrizing Complex Line Integral

  1. Sep 12, 2010 #1
    So this is an ultra basic question, but I'm rusty with parametrization techniques and wanted to make sure I was doing this correctly. Let's say I want to evaluate [tex]\int_{\gamma} z \: dz[/tex] where [tex]\gamma : [a,b]\rightarrow \mathbb{C}[/tex] is some path of integration. Now, I figure I can parametrize the curve and apply the definition of complex integration to arrive at the following: [tex]\gamma(t) = x(t) + iy(t) \quad \text{so} \quad \int_{\gamma} z \: dz = \int_a^b \gamma(t) \gamma(t)' \: dt = \int_a^b (x(t)+iy(t))(x'(t)+iy'(t)) \: dt[/tex] and distribute from there. Again, I know this is a very basic question, and I'm pretty sure it's correct, but it's been a while so I wanted to make sure I wasn't making some silly logical error (quite possible). Thanks.
  2. jcsd
  3. Sep 12, 2010 #2
    So far so good. You are following the definition. You can also calculate this way

    [tex]\int_\gamma f(z)dz[/tex]

    replacing [tex]f(z)[/tex] by [tex]f(\gamma(t))[/tex]
  4. Sep 12, 2010 #3
    Shouldn't that be from t_0 to t_1?
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