representation of delta function I want to prove that 
$\delta(w) = \frac{1}{\pi^2} \int_{- \infty} ^{ \infty} \frac{dy}{y(y-w)}$
Could anyone help? I did the integration in two parts: $w=0$ and $w$ is not zero and I showed that for $w=0$, integral becomes infinite and for $w$ is not equal to zero it becomes zero.  But I don't know why $\frac{1}{\pi^2}$ is present in the question. Could anyone add a better answer rather than mine?

It comes from the Hilbert transform but you have to be very careful about
how you define the integrals. You define:
$$ H(u)(t) = \frac{1}{\pi} {\rm P.V.} \int \frac{u(\tau)}{t-\tau} \; d\tau$$
A remarkable (and non-trivial) identity is that $H(H(u))(s)=-u(s)$ and your expression amounts to evaluating this for $s=0$. The identity $H\circ H=-{\rm Id}$ is valid in $L^p$, $1< p<+\infty$ but as you evaluate at a point, I suspect you need $u$ continuous or better.
