Can anyone simplify the following expression? I guess something from Fourier transform can help:
$$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{i \omega r^{-\gamma}}} \mathrm{d}r$$
where $i$ is the imaginary unit, and $\gamma > 2$.
If you expand the exponential in the integral, only the constant term survives, yielding $R^2/2$, so the limit is $1/2$. This should in fact work for any $\gamma\gt0$, though one might have to argue more carefully in that case why the operations are allowed.