I hope that the answer is not somewhere in the forum. I searched without success. I would like to compute the fourier transform of $f:x \in \mathbb{R} \mapsto (1+x^2)^{-\alpha}$ where $\alpha > 1/2$. Or, if there is no closed form, i would like to obtain approximation of $\mathcal{F}[f](k)$ when $k \to 0$. I know how to deal with the case $\alpha = 1$ using complex integration and inverse fourier formula. But I struggle to generalize this method. Do you have any hint or suggestion of approach? Thanks,
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2Wolframalpha only gives it in terms of modified bessel function of the second kind, as $$\mathcal{F}f=\frac{2^{1-\alpha}|\omega|^{\alpha-\frac12}K_{\alpha-\frac12}(|\omega|)}{\Gamma(\alpha)}$$ (for real $\omega$). – student91 Mar 20 '23 at 15:32
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Thank you, very much. It is enough for me, since the asymptotic $|\omega| \to 0$ is well-known for this function. Do you have an idea of how to derive the identity given by wolframalpha? – jvc Mar 20 '23 at 15:41
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4Confirmed by Basset's integral. – metamorphy Mar 20 '23 at 15:43
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These are called Bessel potentials https://en.wikipedia.org/wiki/Bessel_potential. You can already find several formulas and the asymptotic behavior on the Wikipedia page. You can find some other properties in Stein Singular integrals and differentiability properties of functions (1970), Chapter V, Section 3. In particular, a formula I like to get properties of these functions, is the following integral representation $$ B_s(x) = \frac{\omega_s}{2} \int_0^\infty t^{\frac{d-s}{2}-1}\, e^{-\pi/t}\, e^{-\pi |x|^2 t}\,\mathrm d t $$ where $B_s(x) = \mathcal F((1+|x|^2)^{-s/2})$ and $\omega_s = \frac{2\,\pi^{s/2}}{\Gamma(s/2)}$. To prove it, you can proceed as I did here in the case of $s = d+1$.
At $x=0$, it is convergent if $s>d$, and then $B_s(0) = \frac{\omega_s}{2} \int_0^\infty t^{\frac{d-s}{2}-1}\, e^{-\pi/t}\,\mathrm d t = \frac{\omega_s}{\omega_{s-d}}$.
If $s<d$, it behaves in the same way as $\mathcal F(|x|^{-s}) = \frac{\omega_s}{\omega_{d-s}\,|x|^{d-s}}$.
LL 3.14
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