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Contragenic Functions of Three Variables

Cynthia Alvarez-Peña, R. Michael Porter

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.

Contragenic Functions of Three Variables

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.

Paper Structure

This paper contains 12 sections, 15 theorems, 86 equations, 1 table.

Table of Contents

  1. Introduction
  2. Monogenic functions in $\mathbb R^3$
  3. Notation
  4. Antimonogenics and ambigenics
  5. Homogeneous monogenics
  6. Basis for $\mathcal{M}_2(\mathbb B^3)$
  7. Contragenic functions in $\mathbb R^3$
  8. Existence of contragenics
  9. Basic properties of contragenics
  10. Bergman kernel for $\hbox{Vec}\,\mathcal{M}$
  11. Construction of homogeneous contragenic polynomials
  12. Conclusions

Key Result

Proposition 1.1

A function is left monogenic if and only if it is right monogenic. The set of conjugates of monogenic functions coincides with the set of antimonogenic functions in $\Omega$.

Theorems & Definitions (15)

  • Proposition 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Theorem 3.3
  • Proposition 3.5
  • Proposition 3.6
  • Corollary 3.7
  • ...and 5 more