[Math] Understanding Dirac Delta

dirac deltaintegration

I found this:

here: http://www.nada.kth.se/~annak/diracdelta.pdf on page 2

Can anyone explain how and why all the terms are cancelled in the second step?

Best Answer

Warning: This argument is both flawed and dangerous. It may give "intuition" but the steps are FORMAL, and are not meaningful.

What the author wants you to think is that $f(\infty)H(\infty)- f(-\infty)H(-\infty) = 0\cdot 1 - f(-\infty)\cdot 0 = 0-0 = 0$ and $H=0$ on $(-\infty,0)$ so the integral from $-\infty$ to $\infty$ can be replaced by an integral from $0$ to $\infty$.

The problem with this argument is that the diract delta "function" is not a function. There is no real (or extended real) function that has the properties that the delta "function" exhibits. What we are hiding here is that we are not integrating in the usual way, we are actually integrating with respect to a diract delta measure. The integration by parts formula does not hold in general and it is pure dumb luck that pushing symbols around in this way gives the right answer. Be very careful and take what the author says with great caution.

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