Contour integral solutions of linear differential equations which include a generalization of the Airy integral

TL;DR

This paper introduces two families of contour integral solutions to linear differential equations, generalizing the Airy integral and exploring their properties, including exponential decay and growth behaviors in specific sectors.

Contribution

It presents new families of contour integral solutions that extend the properties of the classical Airy integral, with detailed analysis of their behaviors and relationships.

Findings

Generalization of Airy integral properties

Identification of a subfamily with unique properties

Analysis of three linearly independent solutions

Abstract

The Airy integral is a well-known contour integral solution of Airy's equation which has several applications and which has been used for mathematical illustrations due to its interesting properties. We present and derive properties of two families of contour integral solutions of linear differential equations, where one family includes the Airy integral and Airy's equation, such that the family generalizes known properties of the Airy integral which include exponential decay growth in a certain sector. The second family includes a known example and contains a subfamily with interesting properties where a separate analysis of three pairwise linearly independent contour integral solutions of a particular equation is given.