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I want to show that: $\Delta f = c \delta_0$ where $f(x) = \frac{1}{||x||_2^a}$ and $\delta_0$ is the $n$-dimensional Dirac distribution. $n$ is a natural number and I want to find $a(n)$ so that the above equation is true.

I tried calculating out the Laplace operator of $f$ and I get a lovely function, but I have no idea how to get that to be a Dirac distribution, in fact I think that entire way if going about it is probably wrong. Anyone have any tips or ideas?

Emil
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  • What is $a$?${}$ –  Dec 18 '15 at 01:52
  • Sorry, $a$ is an element out of the real numbers. – Emil Dec 18 '15 at 01:53
  • No need to apologize at all. I somehow missed the $c$ in front of your $\delta_0$ and confused myself. –  Dec 18 '15 at 01:54
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    First I would suggest writing the laplacian in spherical coordinates. Then the easy part is showing that, for $r\neq 0$ $\Delta f=0$ for a certain $a$ which is related to $n$ – lcv Dec 18 '15 at 02:01

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