On the relation between Airy integral and Bessel functions revisited

TL;DR

This paper revisits the mathematical relationship between Airy integrals and Bessel functions, providing a simpler proof to aid physicists in understanding radiation spectra from relativistic particles.

Contribution

It offers a new, simplified proof of the relation between Airy integrals and Bessel functions using Bowman transformation, improving accessibility for physicists.

Findings

Presented a new proof based on Bowman transformation

Clarified the relation for applications in radiation physics

Enhanced understanding of spectral distribution calculations

Abstract

The Airy integral and Bessel functions are of significant in mathematical description of spectral distribution of different types of radiation produced by relativistic charged particles moving in synchrotron and in periodical macro- and micro-structures. A simple proof of the relation between Airy integral and modified Bessel function is of most important for radiation physicists, who are not expert in pure mathematics. In this paper we discuss an old proof proposed by Nicholson, and suggest another simple one based on the Bowman transformation of Bessel differential equation with a complex constant.