On the relation between Airy integral and Bessel functions revisited

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.