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I need to calculate the Fourier transform of this peculiar function - $$f_{s,n} (x) =\left(\frac{x-i}{x+i}\right)^{n}\cdot \frac{1}{(x^2+1)^{s}},$$

where one may assume that $0<s<\frac{1}{2}$ and $n\in \mathbb{Z}$ an integer. For $n=0$ I've done the calculation, based on an integral representation for the modified Bessel function known as Basset's Integral - http://dlmf.nist.gov/10.32#i . I've looked up some formulas, and I suspect that this has something to do with something called Whittaker function. I'm especially interested in bounds for the Fourier transform in the region $0<z<1$.

P.S. I have reason to suspect that at-least for $n>0$, there should be a nice relation between $f_{s,n}$ and $f_{s,n+1}$, maybe just differentiation or something, as there is some representation theory involved here.

Thanks for your time and efforts.

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For those who are interested. Writing the function as - $$(x\pm i )^{2n} \cdot \frac{1}{(x^2+1)^{2n+s}}$$ we see that the second factor can be transformed to Bessel function as the $n=0$ case. The first factor is a polynomial that under Fourier transform becomes differential operator. To get explicit formula, one needs to use connection formulas for the derivative of the Bessel function.

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