On a multiplicative distribution of functions in complex plane.

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (- 1) -power symmetry, in a rich subfamily B of the family of all complex valued functions on a symmetric set G ⊂ 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 G . The announced notions allow a unique decomposition of functions f ∈ 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 · f - 1 , for various f ∈ 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. [ABSTRACT FROM AUTHOR]