Asymptotic behavior of integral with sinc and exponential functions I have an integral of the following formbehavior
$$ \int_0^{\infty}\frac{dk}{2\pi^2}\frac{\sin kr}{kr}k^2\exp\left[ -\frac{1}{2}(k R)^n \right] $$
I need to determine the asymptotic behavior for the integral at large distance $s\equiv r/R\rightarrow \infty$. For $n=1$ and $n=2$, it is very easy to solve these integrals to determine the asymptotics. For example, $n=1$ gives a Lorentzian and $n=2$ a Gaussian. Now the question is whether there is a way to generalize this for some arbitrary $n$? I used Mathematica to compute this for $n=$ and the result was in the form of generalized hypergeometric function and the asymptotic behavior was difficult to deduce.
 A: Here's a partial answer valid for odd $n$.  Even $n$ looks like the more interesting case.
Following the comment by @eyeballfrog let's look at the problem
$$
F(s) = \int_0^\infty \sin(su) f(u) \, du
$$
where $f(u) = u e^{-u^n/2}$.  A `standard' approach to produce an asymptotic series for $F$ would be to apply integration by parts repeatedly to find
$$
F(s) = -\frac{1}{s}\left[\cos(su)f(u)\right]_0^\infty + \frac{1}{s^2}\left[\sin(su)f'(u)\right]_0^\infty+ \frac{1}{s^3}\left[\cos(su)f''(u)\right]_0^\infty - \ldots
$$
with the last term an integral that is asymptotically lower order than the last minus one due to Riemann--Lebesgue lemma.
Due to the behaviour of sines/cosines, even terms (corr. to even derivatives of $f$, and odd powers of $s$) in the above series are possibly nonzero.  If we expand $f$ near $u=0$:
$$
f(u) = u - \frac{u^{n+1}}{2} + \frac{u^{2n+1}}{2^2\cdot 2!} - \frac{u^{3n+1}}{2^3\cdot 3!} + \ldots
$$
we see that if $n$ is even, all even derivatives of $f$ are zero, and so every term in the above expansion is zero.  The integral will be asymptotically smaller than any power of $s$.
If $n$ is odd, the first nonzero term in the expansion of $F$ will be the one corresponding to the $(n+1)$th derivative of $f$, the second will correspond to the $(3n+1)$th derivative, and so on:
$$
F(s) \sim \frac{(n+1)!(-1)^{(n-1)/2}}{2} \frac{1}{s^{n+2}} - \frac{(3n+1)!(-1)^{(3n-1)/2}}{2^3 3!} \frac{1}{s^{3n+2}} + \ldots
$$
