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I have trouble dealing with the confirm of following relationship:

$$ \mathscr{F}({\frac{1}{\sqrt{x^2+y^2}}}) = \frac{1}{\sqrt{k_x^2 + k_y^2}} $$ I have tried transforming to the polar coordinate or using some limit trick.Can anybody help me?Please.

Yunfeng Deng
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    Polar is promising. The usual "trick" is to have $k$ point along the $x$ axis so that $(x,y)\cdot(k_1,k_2)= \vert (x,y)\vert\vert(k_1,k_2)\vert\cos(\theta)$. – podiki Nov 16 '21 at 06:06

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Let $f$ be a function homogeneous of degree $d$ in an $n$-dimensional space. Then its Fourier transform is also homogeneous, but of degree $-(n+d)$: $$ \hat{f}(c\xi) = \int f(x) e^{-ix\cdot c\xi} \, dx = \int f(x) e^{-icx\cdot \xi} \, dx = \int f(y/c) e^{-iy\cdot \xi} \, |c|^{-n} dy \\ = \int c^{-d} f(y) e^{-iy\cdot \xi} \, |c|^{-n} dy = c^{-(n+d)} \int f(y) e^{-iy\cdot \xi} \, dy = c^{-(n+d)} \hat{f}(\xi). $$

With $n=2$ and $d=-1,$ which is your case, we get $$ \hat{f}(c\xi) = c^{-1} \hat{f}(\xi) $$ implying that $\hat{f}(\xi) = C|\xi|^{-1}.$ This verifies your relationship modulo the value of the constant $C$.

md2perpe
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