Computing a Fourier transform and imaginary part dependence

Question

I would like to compute the following Fourier transform: $$\int_{-\infty}^\infty (1 + (x+it)^2)^{-s} e^{i x \xi} dx$$ and to explicitly see the dependence in $s$, in particular the imaginary part. Since the variables are nonreal, I am not so sure what kind of change of variables can be done.

Answer 1

This is expressible with the Bessel function $$K_p(a,b)=\int_{0}^{\infty}r^{p-1}e^{-(ar+\frac{b}{r})}dr.$$ To see this, apply $$\frac{1}{A^s}= \int_{0}^{\infty}e^{-Ar}\frac{r^{s-1}}{\Gamma(s)}dr$$ to $A=1+(x+it)^2$ and then Fubini, landing on a Gaussian integral. Computation goes well, but I am afraid to make mistakes for the exact expression of $a$ and $b$ in terms of $\xi$ and $t.$ I have found $$p=s-\frac{1}{2},\ a=1+3t^2, \ b=\frac{\xi^2}{4}.$$

Note that $K_p(a,b)$ can be expressed in terms of $K_p(t)=K_p(t,t)$ with $t=\sqrt{ab}.$