Partially - conformal mappings in multidimensional complex spaces

TL;DR

This paper extends core concepts of complex variable theory, such as modulus and holomorphic functions, to infinite-dimensional complex spaces, enabling generalization of classical geometric function theory results.

Contribution

It introduces a vector-based generalization of complex concepts and extends key theorems to infinite-dimensional complex spaces.

Findings

Generalization of modulus and argument to vector and infinite-dimensional settings

Extension of classical theorems of geometric function theory to infinite-dimensional spaces

Summarization of known results on functions of class S in infinite-dimensional complex spaces

Abstract

The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and mappings to the case of infinite-dimensional complex spaces. This approach allows us to generalize several well-known results of geometric function theory to the case of infinite-dimensional complex spaces. In particular, the author summarizes a number of known theorems on the theory of functions of class S on infinite-dimensional complex space.