Laplacian of step function

Question

Is the second derivative of the step function equal to zero?

$\partial^2_{\theta}\bigg(\Theta(\theta-\frac{\Pi}{2})-\Theta(\theta-\frac{5 \pi}{4})\bigg)= \:?$

Answer 1

What follows is extremely nonrigurous. By integration of parts, we can 'define' $\frac{d(\delta x)}{dx}$ -

\begin{align} \int \delta (x) dx &= x\delta (x) - \int x\frac{d(\delta x)}{dx}dx \end{align}

As $\delta(x) = 0$ at $x\ne 0$, and $x\delta (x) = 0$ everywhere.

So, we can define $\frac{d(\delta x)}{dx} = \frac{-\delta(x)}{x}$.