Computing with Harmonic Functions

TL;DR

This paper introduces a Mathematica package that enables efficient computation of harmonic functions, including solving classical boundary value problems and performing harmonic analysis in various geometric regions.

Contribution

The paper presents a new software tool that automates complex harmonic function calculations, significantly reducing computational effort and expanding analytical capabilities.

Findings

Exact solutions for Poisson integrals of polynomials

Solutions to Dirichlet, Neumann, and biDirichlet problems in R^n

Tools for harmonic analysis and basis computation

Abstract

This document is the manual for a free Mathematica package for computing with harmonic functions. This package allows the user to make calculations that would take a prohibitive amount of time if done without a computer. For example, the Poisson integral of any polynomial can be computed exactly. This software can find exact solutions to Dirichlet, Neumann, and biDirichlet problems in R^n with polynomial data on balls, ellipsoids, and annular regions. It can also find bases for spaces of spherical harmonics, compute projections onto the harmonic Bergman space, and perform other manipulations with harmonic functions.