This problem is motivated from the physics side. But it does not require any background in physics.
Consider the function $g(u)$
$$\frac{1}{\alpha+\beta\left\{\frac{\sqrt{(x+u)^2-1}}{2x}\left[1-2\theta(x+u)\theta(|x+u|-1)\right]-\frac{\sqrt{(x-u)^2-1}}{2x}\left[2(\theta(x-u)-1)\theta(|x-u|-1)+1\right]\right\}}$$
where $x,\alpha,\beta$ are all positive real numbers. $\theta$ is the Heaviside step function.
How to compute
$$\int_{-\infty}^{\infty}e^{iut}g(u)\,du$$.
I tried to used the Jordan’s lemma but the problem is that the function has Heaviside step function, which may not have an analytical continuation in the upper half complex plane.