Elliptic umbilic representations connected with the caustic

TL;DR

This paper explores the elliptic umbilic canonical integral, providing new integral representations, explicit values in hypergeometric functions, and connections to Airy and Bessel functions, advancing understanding of its mathematical properties.

Contribution

It introduces an absolutely convergent integral representation for the elliptic umbilic and expresses its values using hypergeometric functions, linking it to special functions.

Findings

Derived an absolutely convergent integral representation.

Expressed elliptic umbilic values in terms of 2F2 hypergeometric functions.

Connected elliptic umbilic integrals to Airy and Bessel functions.

Abstract

We investigate the elliptic umbilic canonical integral with an approach based on a series expansion of its initial distribution shifted to the caustic points. An absolutely convergent integral representation for the elliptic umbilic is obtained. Using it, we find the elliptic umbilic particular values in terms of 2F2 hypergeometric functions. We also derive an integral over the product of Gaussian and two Airy functions in terms of Bessel functions of fractional orders. Some other corollaries including 3F2 hypergeometric function special values and the Airy polynomials relations are also discussed.