Fundamental Solution to Laplace equation using inverse Fourier

Question

I'm trying to do the following integral (which is an inverse Fourier transform):

$$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{e^{i (x_1\xi_1 + x_2\xi_2) }} {4\pi^2 (\xi_1^2 +\xi_2^2)}d\xi_1d\xi_2$$

Any help with this? I tried converting to polar but that didn't make my life that much easier.

Ron Gordon · Answer 1 · 2014-04-04T20:50:32.860

0

In polars, the integral is equal to

$$\frac1{4 \pi^2} \int_0^{\infty} dk \, \frac1{k} \, \int_0^{2 \pi} d\theta \, e^{i k r \cos{\theta}} = \frac1{2 \pi} \int_0^{\infty} dk \, \frac{J_0(k r)}{k}$$

You should be able to see that the integral does not converge.

edited Apr 04 '14 at 20:50

answered Apr 04 '14 at 20:37

Ron Gordon

138,521

I'm trying to do the inverse Fourier transform of the function $\frac{1}{4\pi^2(\xi_1^2+\xi_2^2)}$, does it not exist? – user137302 Apr 04 '14 at 20:48
@user137302: Well, one would have to conclude, given the above, no. – Ron Gordon Apr 04 '14 at 20:50
where did the $x_1$ and $x_2$ go in your answer above? – user137302 Apr 04 '14 at 21:02
$$r=\sqrt{x_1^2+x_2^2}$$ – Ron Gordon Apr 04 '14 at 21:02
But you're integrating with respect to the $\xi$'s – user137302 Apr 04 '14 at 21:19
@user137302: I switched to polars and let $k=\sqrt{\xi_1^2+\xi_2^2}$. – Ron Gordon Apr 04 '14 at 21:27

Fundamental Solution to Laplace equation using inverse Fourier

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