1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Complex Integration
  1. Complex integral (Replies: 7)

  2. Complex integrals (Replies: 5)

  3. Complex integration (Replies: 1)

  4. Complex integration (Replies: 3)

  5. Complex integration (Replies: 6)