Inverse Fourier Sine Transform of $\frac{1}{p^2+a^2}$

Question

What is the inverse Fourier sine transform of $\frac{1}{p^2+a^2}$? I don't need just the answer, I need an explanation for its solution. I don't know how to start.

It appeared in one of my past assignments. However, now the deadline is over.

Answer 1

Start with this:

$$\begin{align*}\dfrac{1}{2\pi}\int_{-\infty}^\infty \dfrac{1}{p^2+a^2}\sin(px)\;dp &= \dfrac{1}{2}\left[\dfrac{1}{2\pi i}\int_{-\infty}^\infty \dfrac{e^{ixp}}{(p+ia)(p-ia)}\;dp \\ \quad \quad \quad -\dfrac{1}{2\pi i}\int_{-\infty}^\infty \dfrac{e^{-ixp}}{(p+ia)(p-ia)}\;dp\right]\end{align*}$$

and then perform contour integration to solve the individual integrals on the RHS.

Note that for each contour integration you must consider the two cases of $x >0$ and $x < 0$ and change the closed contour accordingly. The final answer will ultimately be expressed in terms of $|x|$.