Fourier transformation of a function

Question

How to use Fourier transformation to compute $\frac{1}{\pi}$ $\int_{-\infty}^{\infty}$ e$^{\frac{itX}{n}}$$\frac{1}{1+x^2}$dx?

And the pdf of X is f(x)= $\frac{1}{\pi}$$\frac{1}{1+x^2}$

I find a bit confusing to deal with this n?

Answer 1

It is very easy to compute (from definition) the Fourier tramsfom of the fun tion $f(t)=e^{-|t|}$. If you apply Fourier Inversion Theorem you will get the F.T. of $\frac 1 {\pi (1+x^{2})}$ and the answer is $e^{-|t|}$. Once you do this you just change $t$ to $\frac t n$.