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 Variables and Integration

  1. Nov 17, 2004 #1
    How do I solve the following integral using complex variable techniques

    The integral from 0 to infinity of [x^m/(x^2 + 1)^2]; 1<m<3
     
  2. jcsd
  3. Nov 18, 2004 #2

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    You will have to select a contour judiciously. Since m is not generally an integer then z = 0 is a branch point. Choose the branch to extend from 0 to infinity along the postive real axis. For your contour, one part will extend from infinity along the positive real axis and loop arbitrarily close to the origin then continue along the positive real axis paying close attention to the fact that [itex]z = R e^{2 \pi i}[/itex] along that portion of the contour. The rest of your contour circles around from [itex]\infty e^{0i}[/itex] to [itex]\infty e^{2\pi i}[/itex].

    Now just apply the Cauchy Integral Theorem! You'll pick up a residue from the small loop around the origin and the two linear segments have different phases and their sum is proportional to the integral you want to evaluate.
     
  4. Nov 20, 2004 #3
    What about the singularities at z=i and at z=-i?
    Do I make my contour so that these two points are not included?
     
  5. Nov 20, 2004 #4

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    Using the contour I described, those singularities are included. You have to be aware of what is inside and outside the contour and it becomes a simple matter of proper bookkeeping!
     
    Last edited: Nov 20, 2004
  6. Nov 20, 2004 #5
    So let me get this straight, we have a contour that extends from infinity along the positive real axis, loop around as it comes close to the origin, and then what does it do on the negative real axis? Sorry, just trying to get this picture in my head.
     
  7. Nov 20, 2004 #6

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    Okay - step by step: Start at infinity along the positive real axis and go toward the origin. Loop around the origin (tiny loop!) and go back out to infinity (on the [itex]e^{2\pi i}[/itex] side!) then close your loop with a big circle at infinity. We're choosing [itex]z = R e^{2\pi i}[/itex] to be the branch cut.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Complex Variables and Integration
  1. Complex Variables (Replies: 2)

  2. Complex integral (Replies: 6)

  3. Complex integration (Replies: 6)

Loading...