I'm attempting to numerically handle an equation of the following form:
\begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*}
where $f(\omega) = \int_{0.0001}^{2} dk \frac{1}{\sqrt{w^2+k^2}} e^{-\sqrt{w^2+k^2}*2}*\frac{1-(1-(w^2+i*w*0.001)^{-1})}{1+(1-(w^2+i*w*0.001)^{-1})}$
which for $\omega$ in the integration range is bounded in both real and imaginary space between -10 and +10. I would like to solve this integral for $x \in (-500,100)$ but the oscillatory nature means a special technique should be used.
Traditionally for integrals where the oscillation is with a fixed function $e^{I\omega x}$, the method of summing over iterations of integration by parts, namely that:
\begin{equation*}\int_{a}^{b} d\omega e^{i \omega x} f(\omega) = -\frac{i}{x} \sum_{r=0}^{\infty} \left( \frac{i}{x}\right)^r (e^{ixb}f^{(r)}(b) - e^{ixa}f^{(r)}(a))\end{equation*} for $f^{(r)}$ the $r$th derivative with respect to $\omega$ of the function in question.
$f(\omega)$ has two regions with very large derivatives, making the sum over $r$ divergent after 3 iterations. Given that stationary phase is unfeasible, what is the best approach to handle these misbehaving functions?