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Complex Integration

  1. Oct 16, 2007 #1
    1. The problem statement, all variables and given/known data

    Compute the following integrals using the principle value of [tex]z^{i}[/tex]

    [tex]\int z^{i} dz [/tex] where [tex]\gamma_{1}(t)=e^{it}[/tex] and [tex]\frac{-\pi}{2}\leq t \leq \frac{\pi}{2} [/tex]

    [tex]\int z^{i} dz [/tex] where [tex]\gamma_{1}(t)=e^{it}[/tex] and [tex]\frac{\pi}{2}\leq t \leq \frac{3\pi}{2}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    There is a "hint" with the problem that says one of the integrals is easier than the other.
    I don't see why, for part a, I can't use a branch cut along the negative real axis, so that [tex]z^{i}[/tex] will be analytic along the path.
    And for part b, i don't see why I can't simply use a different branch cut, say one along the positive real axis, so that [tex]z^{i}[/tex] will then be analytic along that path.

    Then for each, I can just take the antiderivative:


    and plug in the end points...

    If I do, I get:

    a. [tex]\int z^{i} dz = \frac{i^{i+1}}{i+1}+\frac{(-i)^{i+1}}{i+1} = (\frac{e^{\frac{\pi}{2}}}{2}+\frac{e^{\frac{\pi}{2}}}{2}i)-(-\frac{e^{\frac{\pi}{2}}}{2}-\frac{e^{\frac{\pi}{2}}}{2}i ) = cosh(\frac{\pi}{2})+cosh(\frac{\pi}{2})i[/tex]

    and I will get the same thing for b.

    Is there something I am missing? Do I ned to parameterize the integral or is what I am doing correct?

    Thanks in advance!
    Last edited: Oct 16, 2007
  2. jcsd
  3. Oct 16, 2007 #2
    I'm thinking that because [tex]f(z)=z^{i}[/tex] is entire, and that the region in which the curve lies will be simply connected.... then the anitderiv exists and since [tex]i[/tex] is just a constant, then the primitive of [tex]f(z)[/tex] will be [tex]F(z)=\frac{z^{i+1}}{i+1}[/tex]...

    does any one have any ideas? i'm really stuck here... thanks
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