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From Mathworld, for example, we have the following properties of the Dirac delta:

$x^n\delta^{(n)}(x)=(-1)^n\, n! \, \delta(x)$

$x^2 \, \delta'(x)=0$

So, if $f(x)$ is $C^\infty(R)$, is it correct to guess that, from its Taylor expansion:

$f(x) \, \delta''(x-a)=f(a)\delta''(x-a)-f'(a)\delta'(x-a)+f''(a)\delta(x-a)$?

mattiav27
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    It is important to realize that all the properties of the $\delta$-function (and its derivatives) are obtained when one multiplies it by a test-function $f(x)$ and integrates from $-\infty$ to $+\infty$. The results you cite are therefore merely a sort of short-hand notation. They have been found by partial integration. That is also what you should do, and then things become fairly obvious. Certainly you should not "guess" the result !!! – M. Wind Jun 21 '15 at 16:43
  • I can tell you the result for the left hand side. If we perform partial integration twice in succession (as described above), we get: $f(x) \delta''(x-a)$ = $-f'(x) \delta'(x-a)$ = $f''(x) \delta(x-a)$. – M. Wind Jun 21 '15 at 16:51
  • Having said that, it is certainly permitted to expand the test-function $f(x)$ in a Taylor-series around $x=a$ and then to apply partial integration to each term. Thus I can confirm the correctness of the OP's expansion. [There is only a minor error in the second term on the RHS; the argument should be $x-a$ instead of $x$.] – M. Wind Jun 21 '15 at 17:24
  • While the "bottom line" is as @M.Wind says, that $x^2$ times a distribution $u$ really means $(x^2 u)(f)=u(x^2f)$ for test function $f$, this very thing does make sense of "multiplication of distributions by smooth functions". That is, those left-hand sides really do have sense as distributions, as does the right-hand side... but using the argument "x" potentially confuses things, yes! Might be better to write just "$f$" rather than $f(x)$, and $\delta_a$ rather than $\delta(x-a)$. The usual notation for $x\to x^2$ as being just "$x^2$" may be inescapable... – paul garrett Jun 21 '15 at 17:34
  • The OP should keep in mind that while his expansion is okay, the first and second term on the RHS do not contribute anything ! – M. Wind Jun 21 '15 at 17:34

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