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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

    [tex]p(x)=\frac{1}{\pi}\frac{1}{1+x^2}[/tex]


    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]u=\frac{1}{1+x^2}[/tex]
    [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

    SteamKing

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    You haven't written the correct ∫ v du term any way.
     
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