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I know the result of one dimensional case.

However, I can't find out what it is for the multidimensional case, i.e. the following integrals. $$ \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\mathbb{R}^n} e^{-a|x|} \,e^{-i \xi x}\,\mathrm d x $$ I tried to use n-dimensional spherical coordinate, so I arrived to the following $$ \frac{1}{(2\pi)^{n/2}} \int_0^\infty \int_0^\pi x^{n-1} \sin^{n-2}(\theta)\, e^{-ax-i|\xi|x\cos(\theta)} \,\mathrm d\Omega_{n-2}\,\mathrm d\theta\,\mathrm d x $$

where $\mathrm d\Omega_{n-2}$ is solid angle for $S^{n-2}$

But, I cannot go further. I can't find out how to do integration. How can I do the integration?

LL 3.14
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김동엽
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  • I don't have time to write an answer right now, but check Theorem 3 of this paper: https://www.math.arizona.edu/~faris/methodsweb/hankel.pdf – md2perpe Oct 12 '21 at 08:30
  • Thank you a lot, I didn't know about Hankel transformation, so It's really interesting. But, still, I don't know well about the property of Bessel function, and I still have difficulty on deriving the result I want – 김동엽 Oct 12 '21 at 15:11
  • The integral isn't trivial. – md2perpe Oct 12 '21 at 15:39

1 Answers1

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As proved in the answer to this question, the answer is with your conventions $$ \boxed{\mathcal F(e^{-a\,|x|}) = \frac{2\,a \,(2\,\pi)^{n/2}}{\omega_{n+1}\,(a^2 + |x|^2)^{(n+1)/2}}} $$ where $\omega_{d} = \frac{2\pi^{d/2}}{\Gamma(d/2)}$.

LL 3.14
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