Pearcey integral explained

In mathematics, the Pearcey integral is defined as^[1]

\operatorname{Pe}(x,y) = \int ^{inf ty} _{-inf ty} \exp(i(t ^{4+xt} ^{2+yt))} dt.

The Pearcey integral is a class of canonical diffraction integrals, often used in wave propagation and optical diffraction problems^[2] The first numerical evaluation of this integral was performed by Trevor Pearcey using the quadrature formula.^{[3] [4]} In optics, the Pearcey integral can be used to model diffraction effects at a cusp caustic, which corresponds to the boundary between two regions of geometric optics: on one side, each point is contained in three light rays; on the other side, each point is contained in one light ray.

References

1. Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, NIST Handbook of Mathematical Functions, p. 777, Cambridge, 2010
2. Book: 10.1017/CBO9780511753626. [{{Google books|title=Hadamard Expansions and Hyperasymptotic Evaluation|8WggWK8AecIC|page=207|plainurl=yes}} Hadamard Expansions and Hyperasymptotic Evaluation]. 2011. Paris. R. B.. 9781107002586.
3. 10.1080/14786444608561335. XXXI. The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. 1946. Pearcey. T.. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 37. 268. 311–317.
4. 10.1090/mcom/3164. Analytic formulas for the evaluation of the Pearcey integral. 2016. López. José L.. Pagola. Pedro J.. Mathematics of Computation. 86. 307. 2399–2407. 1601.03615.