Characterization of polyanalytic functions by meromorphic extensions into chains of circles

TL;DR

This paper characterizes polyanalytic functions using meromorphic extensions into chains of circles, providing a new criterion based on the behavior of functions inside these circles.

Contribution

It introduces a novel characterization of polyanalytic functions through meromorphic extensions with poles at centers of circles in a smooth family.

Findings

Polyanalytic functions are characterized by their meromorphic extensions.

Extension property holds for functions with poles only at circle centers.

Provides a new tool for identifying polyanalytic functions in complex analysis.

Abstract

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.