- #1

fluidistic

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## Homework Statement

I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable [itex]f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2}[/itex] which is basically a kind of Lorentzian.

## Homework Equations

The characteristic function is simply a Fourier transform, namely [itex]\phi _X (k)= \int _{-\infty } ^{\infty } \frac{e^{ikx}}{x^2+c^2}dx[/itex].

## The Attempt at a Solution

My problem resides in evaluation the integral. If there wasn't that exponentional in the numerator, I'd get an arctangent but unfortunately I have a complex exponentional there.

Is [itex]\phi _X (k)=\frac{ce^{ik}}{\pi} \int _{-\infty}^{\infty} \frac{e^x}{x^2+c^2}dx[/itex] valid and a good start?

Edit: Hmm no ! Very bad. This would make the integral not convergent... How is that possible? The range of x is -infinity to infinity!

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