Finding a characteristic function (integral help)

[mjordan2nd]

Homework Statement

I'm looking to find the characteristic function of

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

Homework 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.[/B]

 

[SteamKing]

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