Complex-Valued Functions with NonDegenerate Groups of Regular Points

Володимир Володимирович Павленков

Abstract


Complex-valued functions with nondegenerate groups of regular points are studied in the paper. A class of functions f, which takes value on the complex plain and for which the limit  \[k_{f}\left ( \lambda \right )=\lim_{x\to 0}\frac{f\left ( \lambda x \right )}{f\left ( x \right )}\] exists, and is nonzero and finite for some  \[\lambda\] from subset of positive real numbers is considered. It was received that this subset is multiplicative group, and it is called the group of regular points. Functions with nondegenerate groups of regular points generalize the class of RV functions. The corresponding limit functions are defined for complex-valued functions with nondegenerate groups of regular points. Factorization representations for this limit functions are obtained. It was shown, that for complex-valued function with nondegenerate group of regular points it limit function can be represented as a product of power function and periodic function of logarithm argument. Similar results are known for real valued functions with nondegenerate groups of regular points. Obtained results are generalized and complemented corresponding results from real valued situation. Some well-known theorems from RV functions theory can be covered from this general approach.

Keywords


RV function; ORV function; Group of regular points

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DOI: https://doi.org/10.20535/1810-0546.2014.4.28215

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