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[Introduction to Partial Differential Equations - Gerald B. Folland, chapter 2, section C, question 6]

Show that $$\frac 1{4\pi |r|} e^{-c|r|}$$ is a fundamental solution for $$-\Delta+c^2\qquad (c\in \mathbb C)$$ on $\mathbb R^3$.

I have been able to show that it is indeed a solution, i.e., $$\Delta F = c^2F$$ But, I can't figure out how to show that it is a fundamental solution. Please help me in that.

Also, the way I proved that it is a solution, it took me pages of hardwork - will it be possible to simplify the calculations somehow?

Note that I know what a Fundamental Solution is. I just don't see any way to complete the required calculation in finite amount of time :)

Note that here $r=\sqrt{x^2+y^2+z^2}$.

Sayan Dutta
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  • What needs to be established in order for this to be a Fundamental solution? – user619894 Oct 05 '23 at 13:50
  • See Fundamental solution. – Gonçalo Oct 05 '23 at 15:14
  • "I have been able to show that it is indeed a solution."

    A solution to what?

     – Didier Oct 05 '23 at 15:36
  • @Didier please check the edits – Sayan Dutta Oct 05 '23 at 16:02
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    Fundamental solution doesn't mean "it's a solution of the homogeneous equation $(-\Delta+c^2)F=0$ that's also fundamental" for some meaning of the word fundamental. It means instead "it's a solution of the inhomogeneous impulse equation $(-\Delta +c^2)F=\delta$. If you have showed that it is a solution to this equation, then you're done. – ziggurism Oct 05 '23 at 20:03
  • You can just use the Fourier transform as in here https://math.stackexchange.com/questions/4220006/computing-the-fourier-transform-of-exponential-decay-in-mathbbr2/4220125#4220125 (replacing $d+1$ by $2$ in the proof). Or use the result and compute the convolution of exponentials ... – LL 3.14 Oct 05 '23 at 23:07

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