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skate_nerd
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Homework Statement
Prove this theorem regarding a property of the Dirac Delta Function:
$$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$
(by using integration by parts)
Homework Equations
We know that δ(x) can be defined as
$$\lim_{\sigma\to\infty}(\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{x^2}{2\sigma^2}})$$
and another proper of the Dirac Delta Function is
$$\int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a)$$
Also, δ(x-a)=0 when x is not equal to a, and δ(x-a) is undefined when x=a. (Not really sure how to interpret the second part of that statement quite yet, I understand that it is because the two sided limit of σ->0 doesn't exist, but I don't see how to picture that graphically)
The Attempt at a Solution
Starting with
$$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$
I tried integration by parts, taking u=f(x) so u'=f'(x), and then v'=δ'(x-a) so v=δ(x-a), translating our original integral to
$$f(x)\delta(x-a) (from -\infty to \infty)-\int_{-\infty}^{\infty}f'(x)\delta(x-a)dx$$
I'm really not sure where to go next. I don't see what else I can do with evaluating the first part from negative infinity to infinity, and I don't see anything more I can do with the remaining integral. All in all I don't see how integration by parts was helpful at all.
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