2D Fourier transform of $1/(x^2-y^2+q)$ How can I calculate the following 2D Fourier integral:
$$
\iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2+q} {\rm d}x\,{\rm d}y,
$$
where $q$ is a complex number?
If there was a "+" sign in the denominator: $(x^2+y^2+q)^{-1}$, I'd use polar coordinates to get to the Hankel transform of a simple function $(r^2+q)^{-1}$, that can be calculated in terms of modified Bessel function. But I've no idea how to deal with the $(x^2-y^2+q)^{-1}$ function.
Update:
I am also interested in calculating even more general integral
$$
\iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2 + sy+q} {\rm d}x\,{\rm d}y,
$$
where both $s$ and $q$ are complex numbers (for example $s=-2{\rm i}$, $q=4-{\rm i}$).
I have reasons to believe that this integral can be calculated in a closed form using Bessel $J_0$ function.
 A: Let us consider a function
$$
f(x,y) = \frac{1}{x^2+ay^2+q}.
$$
If $a$ and $b$ is a positive real values then the Fourier transform is
$$
\hat{f}(k_x,k_y) = \frac{2 \pi}{\sqrt{a}}  K_0\left(\sqrt{q (k_x^2+k_y^2/a)}\right).
$$
It can be easily derived as in your link. Now you can try to analytically continue the function to other values of $a$ and $q$. I.e. just substitute desired values of $a$ and $q$ to the formula above (and maybe choose the right branch of square roots). I numerically verify it and it works! See my post on mathematica SE for details.
A: I'll assume the exponential is supposed to be $\exp\left[-i\left(ax+by\right)\right]$ instead of $\exp\left[+i\left(ax+by\right)\right]$.
Here's what I would try. Write this as:
$$
\int_{-\infty}^{\infty} dy \ e^{-iby} \left[\int_{-\infty}^{\infty} dx \ e^{-iax} \frac{1}{x^2 + \left(q-y^2\right)}\right]
$$
The expression in brackets is calculated here (or Google a bit for "lorentzian fourier transform"), though you'll have to be careful since you said $q$ is complex.
Once you've evaluated the integral in brackets, do the $y$ integral.
