Quadratic stationary endpoint with quadratically vanishing amplitude (${\int }_0^b{e}^{ix{t}^2}a(t)dt,a(t)=c{t}^2+O(t^4)$)

The stationary endpoint contributes $c{e}^{3i\pi /4}\sqrt{\pi }/(4x^{3/2})$ by method of steepest descent along $t=e^{i\pi /4}r$. Its ordinary $x^{-1/2}$ term vanishes because $a(0)=0$. A nonstationary upper endpoint with $a(b)\ne 0$ contributes $a(b)e^{ix{b}^2}/(2ibx)$ and therefore dominates the stationary contribution. The stationary phase method must retain both this boundary term and the first nonzero local-amplitude term.