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La formule de Cauchy dans les algèbres

Pierre Bonneau, Emmanuel Mazzilli

TL;DR

This paper develops a framework to generalize the Cauchy representation formula to finite-dimensional algebras by introducing first-order Cauchy (CR) conditions and constructing a fundamental solution for the Cauchy–Riemann operator. It analyzes real, constant-coefficient, and variable-coefficient CR systems, exploring both commutative and noncommutative algebras and providing explicit determinant-based ellipticity conditions that govern the existence of integral representations. Through examples in dimensions 2 and 4 (including matrix and Clifford algebras) and Hartogs-type extension results, it delineates when Cauchy kernels exist and when they do not. The work also establishes Hartogs-type and Liouville-type results for algebras with admissible CR conditions and presents an annexe of exotic algebras, illustrating the limitations and reach of Cauchy formulas in nonstandard settings.

Abstract

We Study versions of Cauchy formula in more general algebras than the complex case.

La formule de Cauchy dans les algèbres

TL;DR

This paper develops a framework to generalize the Cauchy representation formula to finite-dimensional algebras by introducing first-order Cauchy (CR) conditions and constructing a fundamental solution for the Cauchy–Riemann operator. It analyzes real, constant-coefficient, and variable-coefficient CR systems, exploring both commutative and noncommutative algebras and providing explicit determinant-based ellipticity conditions that govern the existence of integral representations. Through examples in dimensions 2 and 4 (including matrix and Clifford algebras) and Hartogs-type extension results, it delineates when Cauchy kernels exist and when they do not. The work also establishes Hartogs-type and Liouville-type results for algebras with admissible CR conditions and presents an annexe of exotic algebras, illustrating the limitations and reach of Cauchy formulas in nonstandard settings.

Abstract

We Study versions of Cauchy formula in more general algebras than the complex case.
Paper Structure (9 sections, 1 theorem, 233 equations)

This paper contains 9 sections, 1 theorem, 233 equations.

Table of Contents

  1. Introduction.
  2. Le cas des fonctions réelles de variables réelles.
  3. Conditions de Cauchy à coefficients constants.
  4. Le cas particulier d'une algèbre commutative.
  5. Un autre exemple dans une algèbre de dimension 4 et quelques applications
  6. Théorème de Hartogs
  7. Solution de Cauchy à support compact
  8. Conditions de Cauchy à coefficients variables.
  9. Annexe : quelques exemples exotiques

Key Result

Proposition 3.3

Soit $A$ une algèbre unitaire, associative, et des conditions de Cauchy admissibles sur $A$. Alors les fonctions de $A$ dans $A$ entières et bornées sur $A$ sont constantes.

Theorems & Definitions (3)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3