On a multiplicative distribution of functions in complex plane

Abstract

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (-1)-power symmetry, in a rich subfamily mathcal {B} of the family of all complex valued functions on a symmetric set mathcal {G}subset mathbb {C}. The first one defines the values f(-z) as identical with f(z) and the other as the inverse of the f(z) in mathcal {G}. The announced notions allow a unique decomposition of functions fin mathcal {B} into a product of two factors f_{1},f_{-1} having the (1)- and the (-1)-power symmetry property, respectively. In the paper there are also given examples of such partitions f=f_{1}cdot f_{-1}, for various fin mathcal {B}, a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition.