Oscillatory integral

In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.

An oscillatory integral ${\displaystyle f(x)}$ is written formally as

where ${\displaystyle \phi (x,\xi )}$ and ${\displaystyle a(x,\xi )}$ are functions defined on ${\displaystyle \mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}}$ with the following properties.

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