Q:

Integral of Lorentzian type with trigonometric function

Consider the following Riemann integral
$$
\int_0^\infty \mathrm{d}x \frac{\alpha^2}{(x-x_0)^2+\alpha^2} \frac{\sin\left[{\left(x - x_1\right) t }\right]}{x-x_1}
$$
with the displacements $x_0,x_1 \in \mathbb{R}$ and broadening $\alpha \in \mathbb{R}$. The (claimed) solution should be
$$
\pi \left[\frac{\alpha^2}{\delta^2 + \alpha^2} + e^{-\alpha t}\frac{\delta^2 \sin{(\delta t)}- \alpha^2\cos{(\delta t)}}{\delta^2 + \alpha^2} \frac{}{} \right]
$$
with $\delta = x_0 - x_1$. Is an approximation involved, and how do I evaluate this integral? 

A:

Start from a simplified form involving fewer constants, to grasp the idea; we need
$$
\int_{-\infty}^{+\infty} \left(\frac{1}{1+t^2}\right) \frac{\sin t}{t}dt;
$$
notice that the integrand is the product of two Fourier transforms:
$$
\int_{-\infty}^{+\infty}\frac{1}{2}\chi_{[-1,1]}(x)e^{-itx}dx=\frac{\sin t}{t}
$$
where $\chi$ denotes the characteristic function,
and
$$
\int_{-\infty}^{+\infty}\frac{1}{2}e^{-|x|}e^{-ixt}dx=
\Re\int_{0}^{+\infty}e^{-x(1+it)}dx=
\Re \frac{1}{1+it}=\frac{1}{1+t^2}. 
$$
Now, the $L^2$ product of the Fourier transforms is $2\pi$ times the $L^2$ product of the original functions, i.e.:
$$
\int_{-\infty}^{+\infty} \left(\frac{1}{1+t^2}\right) \frac{\sin t}{t}dt=
2\pi\int_{-\infty}^{+\infty}\frac{1}{2}\chi_{[-1,1]}(x)\frac{1}{2}e^{-|x|}dx={\pi}\int_{0}^1e^{-x}dx={\pi(1-e^{-1})}. 
$$
Now, we try to put the constants in this framework. 
Explicitly: for $x=x_0+s$, denoting $\delta\equiv x_0-x_1$, we get
$$
\int_{-\infty}^{+\infty}\frac{\alpha^2}{s^2+\alpha^2}\frac{\sin((\delta+s)t)}{\delta+s}ds;
$$
now,
$$
\int_{-\infty}^{+\infty}\frac{\alpha e^{-\alpha|v|}}{2}e^{-iyv}dv=\frac{\alpha^2}{\alpha^2+y^2}
$$
whereas
$$
\int_{-\infty}^{+\infty}e^{-ivy}e^{-iv\delta }\frac{1}{2}\chi_{[-1,1]}\left(v/t\right)dv=\frac{\sin((\delta+y)t)}{\delta+y},
$$
by the above mentioned theorem:
$$
\int_{-\infty}^{+\infty}\frac{\alpha^2}{s^2+\alpha^2}\frac{\sin((\delta+s)t)}{\delta+s}ds=
2\pi
\int_{-\infty}^{+\infty}
\frac{\alpha e^{-\alpha|v|}}{2}
e^{-iv\delta }\frac{1}{2}\chi_{[-1,1]}\left(v/t\right)
dv. 
$$
This integral is a bit lengthy, but easy to compute explicitly, yielding precisely:
$$
\int_{-\infty}^{+\infty}\frac{\alpha^2}{s^2+\alpha^2}\frac{\sin((\delta+s)t)}{\delta+s}ds=
\pi \left[
\frac{\alpha^2}{\alpha^2+\delta^2}+e^{-\alpha t}\frac{\alpha\delta \sin(\delta t) - \alpha^2 \cos(\delta t)}{\alpha^2+\delta^2}
\right]. 
$$
[Side note: since the integrand falls off as $1/x^3$, the Lebesgue integral works just as fine in this case.]