A complex Lie algebra of rotationally symmetric operators and their harmonics

TL;DR

This paper explores solutions to rotationally symmetric PDEs in the complex plane derived from a four-dimensional complex Lie algebra, revealing their series representations and connections to classical harmonic functions.

Contribution

It introduces a new class of generalized harmonic functions linked to a specific complex Lie algebra, extending classical PDE solutions.

Findings

Solutions have canonical series representations.

Retrieves classical Laplace and Helmholtz solutions as special cases.

Expresses solutions in confluent hypergeometric functions.

Abstract

We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which such generalised harmonic functions derive. We show that every one of these solutions have a canonical series representation and retrieve those obtained in the case of Laplace and Helmholtz equation. These sums are given in confluent hypergeometric terms that asymptotically correspond to the complex exponential function.