Joe Kamimoto, Hiromichi Mizuno
The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result is also given.
This paper contains 5 sections, 8 theorems, 55 equations.
Theorem 1.1
(i) If $\alpha>0$ is a rational number, then for any positive integer $N$, where $\psi_N(t)$ is a $C^{\lceil \frac{N+1}{\alpha}\rceil-1}$ function on ${\mathbb R}_+$ and for $n\in{\mathbb N}$, where $\Gamma$ means the Gamma function. (ii) If $\alpha>0$ is not a rational number, then for any positive integer $N$, where $\phi_N(t)$ is a $C^{\lceil \frac{N+1}{\alpha}\rceil-1}$ function on ${\mat
