# Plancherel formula and integral computing

1. Dec 9, 2011

### rayman123

1) compute the Fourier transform of the function $$\frac{x}{1+x^2}$$

2) compute the integral $$\int_{-\infty}^{\infty}\frac{x^2}{(1+x^2)^2}dx$$

1) I can write my function as $$x \cdot \frac{1}{1+x^2}$$ and by using the formula

we let $$f(x)=\frac{1}{1+x^2}$$

$$\mathcal{F}[xf(x)]=i(f^{\wedge})^{'}(\xi)=-\pi ie^{-|x|}$$

which finally gives gives

$$\Bigl(\frac{x}{1+x^2}\Bigr)^{\wedge}=\begin{cases} -\pi ie^{-|\xi|} \therefore \xi>0 \\ \pi ie^{-|\xi|} \therefore \xi<0\end{cases}$$

2) this one is a bit tricker and somehow it seems like I am on the correct track except for the sign I get....

I use Plancherel formula for Fourier transform to solve this integral, namely
$$||f^{\wedge}||^2=2 \pi ||f||^2$$

and we have

$$\int_{-\infty}^{\infty}\frac{x^2}{(1+x^2)^2}dx=||\frac{x}{1+x^2}||^2=\frac{1}{2 \pi}||\overbrace{\frac{x}{1+x^2}}^{\wedge}||^2=$$

and we know from the part 1) that

$$\Bigl(\frac{x}{1+x^2}\Bigr)^{\wedge}=\begin{cases} -\pi ie^{-|\xi|} \therefore \xi>0 \\ \pi ie^{-|\xi|} \therefore \xi<0\end{cases}$$

then our integral will be

$$\int_{-\infty}^{\infty}\frac{x^2}{(1+x^2)^2}dx=\frac{1}{2\pi}\int_{-\infty}^{\infty}-\pi^2 e^{-2|\xi|}d\xi=-\frac{\pi}{2}\cdot 2 \int_{0}^{\infty}e^{-2\xi}d\xi=-\pi \int_{0}^{\infty}e^{-2\xi}d\xi=$$

$$\frac{\pi}{2}\Bigl[e^{-2\xi}\Bigr]_{0}^{\infty}=-\frac{\pi}{2}$$

the answer should be $$\frac{\pi}{2}$$ where do I make mistake?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Dec 10, 2011

### susskind_leon

You already wrote down the solution.
||.||^2 means absolute value squared, so the integrand is positive.

3. Dec 10, 2011

### rayman123

yeah, you are right :) thank you