Contragenic Functions of Three Variables

TL;DR

This paper investigates contragenic functions in three variables, revealing their unique properties and providing tools like a Bergman kernel and orthonormal basis, highlighting differences from complex and quaternionic harmonic functions.

Contribution

It introduces the concept of contragenic functions in three variables and develops foundational tools such as a Bergman kernel and an orthonormal basis.

Findings

Contragenic functions are orthogonal to sums of monogenic and antimonogenic functions.

A Bergman kernel for contragenic functions is derived.

An orthonormal basis for contragenic functions in the ball B^3 is constructed.

Abstract

It is shown that harmonic functions from a simply connected domain in R^3 to R^3 cannot always be expressed as a sum of a monogenic (hyperholomorphic) function and an antimonogenic function, in contrast to the situation for complex numbers or quaternions. Harmonic functions orthogonal in L_2 to all such sums are termed "contragenic" and their properties are studied. A "Bergman kernel" and is derived, whose corresponding operator vanishes precisely on the contragenic functions. A graded orthonormal basis for the contragenic function in the ball B^3 is given.