Let$$K_{a,b}(x)=\int_{0}^{\infty}{\psi({\xi})\xi^{-b}e^{ix\xi\pm \xi^{a}}d\xi}\quad a,b>0$$ where $\psi\in C^{\infty}$, equals to $0$ when $\xi<\frac{1}{2}$, and equals to $1$ when $\xi>1$.
As far as I know, when $0<a<1$, $a=1$,and $a>1$,the behaviors of $K_{a,b}$ are quite different (for proper choice of $b$, the singularity lies in $x=0$,$|x|=1$, $x=\infty$ respectively). I want to know the reason behind this and I also want to know how to evaluate such kind of integrals.