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This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as

$$-f'(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$$

where $\delta$ is the Dirac delta. Suppose that I want to integrate then

$$-\int_x^\infty dx' f'(x'-a)=\int_x^\infty dx' \sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$$

where if $C_0=1$ and $f(\infty)=0$ we get

$$f(x-a)=\theta(a-x)-\sum_{n=1}\frac{C_n}{n!}\delta^{(n-1)}(x-a)$$

My questions are:

  1. Where does the minus sign come from? I understand that $\int dx \delta^{(n)}f(x)=(-1)^nf^{(n)}(0)$, but this seems a little different.

  2. Is this kind of treatment legitimate, or is it more of an abuse of the delta function?

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