The integral in the question $$\int_{|\mathbf{x}|<R} d^n \mathbf x \frac{e^{i \mathbf k\cdot \mathbf x}}{|\mathbf x|^2+a^2}$$ arises quite many times when I study physics. ($a, R, \mathbf k$ are constants. I am especially interested in the asymptotic behavior of the integral as $R \to \infty$ or $a \to 0$ with fixed $\mathbf k$.) Since this integral is highly oscillatory, it seems hard to estimate the integral.
I have heard that harmonic analysis deals with the Fourier transform, so it might be related to this integral. What mathematics do I need to study to better understand the above integral? (I have a relatively strong background in mathematics. I know Lebesgue integration and basic functional analysis.)
Changing to hyperspherical coordinates gives $$I(R, a) = \int_{x < R} \frac {e^{i \boldsymbol k \cdot \boldsymbol x}} {x^2 + a^2} d\boldsymbol x = S_{n - 2} \int_0^R \int_0^\pi \frac {e^{i k r \cos \theta}} {r^2 + a^2} r^{n - 1} \sin^{n - 2} \theta \, d\theta dr = \\ (2 \pi)^{n/2} k^{1 - n/2} \int_0^R \frac {r^{n/2}} {r^2 + a^2} J_{n/2 - 1}(k r) dr.$$ If $n \geq 3$, the asymptotic behavior for small $a$ is just given by the corresponding limit: $$\lim_{a \to 0} I(R, a) = I(R, 0) = \frac {\pi^{n/2} R^{\hspace {01px} n - 2}} {\left( \frac n 2 - 1 \right) \Gamma {\left( \frac n 2 \right)}} \hspace {1px} {_1 \hspace {-1.5px} F_2} {\left( \frac n 2 - 1; \frac n 2, \frac n 2; -\frac {(k R)^2} 4 \right)}.$$ For $r \to \infty$, $$\frac {r^{n/2}} {r^2 + a^2} J_{n/2 - 1}(k r) = -\sqrt {\frac 2 {\pi k}} \, r^{(n - 5)/2} \sin \left( k r - \frac {\pi (n + 1)} 4 \right) + o(r^{(n - 5)/2}).$$ If $n \geq 6$, it appears that the integral of the remainder term is $o(R^{(n - 5)/2})$ (I haven't proved this rigorously). Applying integration by parts to the leading term gives $$I(R, a) = \frac 1 \pi \left( \frac {2 \pi} k \right)^{(n + 1)/2} R^{(n - 5)/2} \cos \left( k R - \frac {\pi (n + 1)} 4 \right) + o(R^{(n - 5)/2})$$ for $R \to \infty$.