We study the arithmetical function ``exponent (order) of an integer modulo m" which is here shortly named ``period" of m. A method is developed, named ``separation of parameters", that leads to analytic representation of the function period. Though Bessel functions have dominant role, other special functions are also applicable. The most promising result is derived by making use of Mukisi\'nski's concept of distributions. The developed method, besides its general nature, makes it possible to study computability of arithmetical function period by means of analytic procedures.