1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Finding a characteristic function (integral help)

  1. Mar 17, 2015 #1
    1. The problem statement, all variables and given/known data

    I'm looking to find the characteristic function of


    2. Relevant equations

    The characteristic function is defined as

    [tex]\int_{-\infty}^{\infty} e^{ikx}p(x)dx[/tex]

    3. The attempt at a solution

    I attempted to solve this using integration by parts. I get

    [tex]du = -\frac{2x}{(1+x^2)^2} dx[/tex]
    [tex] v = \frac{1}{ik}e^{ikx}[/tex]
    [tex] dv = e^{ikx}dx[/tex]

    This gives me

    [tex] \phi (k) = \frac{e^{ikx}}{ik(1+x^2)} |_{-\infty}^{\infty} + \int_{-\infty}^{\infty} \frac{2xe^{ikx}}{ik(1+x^2)} dx [/tex]

    I'm a little stuck here. My term on the left seems to diverge, and I'm not particularly sure about how to handle my term on the right. For the term on the right, setting [itex]u=1+x^2[/itex] seem to make both integral limits [itex]\infty[/itex], so that doesn't seem right.
  2. jcsd
  3. Mar 17, 2015 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    You haven't written the correct ∫ v du term any way.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted