Weighted stationary phase of higher orders
  An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory. 
Mark McKee, Haiwei Sun and Yangbo Ye
math.CA math.NT