Bicomplex polyharmonicity and polyholomorphy

TL;DR

This paper explores bicomplex harmonic and polyharmonic functions, revealing limitations of classical results and introducing new classes of bicomplex holomorphic functions with comprehensive characterizations.

Contribution

It provides a complete characterization of bc-harmonic functions as hyperbolic real parts of bc-holomorphic functions and extends classical harmonic theory to bicomplex polyharmonic functions.

Findings

Classical harmonic function results do not directly extend to bicomplex harmonic functions.

Characterization of bc-harmonic functions as hyperbolic real parts of specific bc-holomorphic functions.

Introduction of bc-polyholomorphic functions and their relation to bicomplex polyharmonic functions.

Abstract

In this paper, we are concerned with the bicomplex analog of the well-known result asserting that real-valued harmonic functions, on simply connected domains, are the real parts of holomorphic functions. We show that this assertion, word for word, fails for bc-harmonic functions and we provide a complete characterization of bc-harmonic functions that are the hyperbolic real parts of a specific kind of bc-holomorphic functions. Moreover, we extend the result to bicomplex polyharmonic functions, which implies the introduction of specific classes of bc-polyholomorphic functions.