In Wikipedia, it says The Dirac delta function is homogeneous of degree −1, with the following formula:
However, I can not understand why the last equality is true. Can someone show me the detailed computation?
Thanks!
2026-04-04 12:30:24.1775305824
Homogeneous distribution
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After some search on the internet, I found the following explanation, which I think can solve my puzzle:
$\int f(x)\delta(x) dx = \int f(0)δ(x) dx = f(0) \int δ(x) dx = f(0)$, since $f(x)δ(x)$ and $f(0)δ(x)$ are the same since they are both zero for every $x \neq 0$, and $\int \delta (x)= 1$ by the property of the Dirac delta function.