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A Riemann Boundary Value Problem in a Two-Dimensional Commutative Associative Banach Algebra

S. A. Plaksa, R. Pukhtaievych

Abstract

We consider a Riemann boundary value problem for monogenic functions in a two-dimensional commutative associative Banach algebra. We prove theorems on the existence of a solution to this problem under different assumptions on the coefficient and free term of the problem, and give an explicit formula for the solution.

A Riemann Boundary Value Problem in a Two-Dimensional Commutative Associative Banach Algebra

Abstract

We consider a Riemann boundary value problem for monogenic functions in a two-dimensional commutative associative Banach algebra. We prove theorems on the existence of a solution to this problem under different assumptions on the coefficient and free term of the problem, and give an explicit formula for the solution.
Paper Structure (5 sections, 14 theorems, 37 equations)

This paper contains 5 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Preliminaries on Monogenic Functions
  4. Sokhotski-Plemelj Formulas
  5. Riemann Boundary Value Problems for Monogenic Functions

Key Result

Theorem 1

Let a domain $\Omega \subset E$ and $\partial \Omega$ be a rectifiable Jordan curve. If a function $\Phi: \overline{\Omega} \to \mathbb{B}$ is monogenic in a domain $\Omega$ and continuous in $\overline{\Omega}$, then

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1: Cauchy's Integral Theorem
  • Theorem 2: Morera's Theorem
  • Theorem 3: Cauchy's Integral Formula
  • Theorem 4: Taylor's Theorem
  • Theorem 5: Liouville's Theorem
  • Theorem 6: Extended Liouville's Theorem
  • Lemma 1: Principle of Monogenic Continuation
  • Theorem 7: Logarithmic Residue Theorem
  • Theorem 8
  • ...and 5 more