On the Cauchy Integral and Jump Decomposition

TL;DR

This paper discusses the Cauchy integral and jump decomposition, emphasizing their significance in geometric function theory, especially over quasicircles, and explores related concepts like Faber series.

Contribution

It provides a detailed overview of the Cauchy type integral and jump decomposition, contextualizing their roles in geometric function theory and quasicircle analysis.

Findings

Clarifies the properties of the Cauchy integral and jump decomposition.

Highlights the importance of these integrals in geometric function theory.

Connects Faber series to the jump problem in the context of quasicircles.

Abstract

Following "Boundary Value Problems" by Gakhov, we present basic details of the Cauchy Type Integral and its Jump Decomposition. We also contextualize its place and importance in Geometric Function Theory, and efforts to define these integrals over Quasicircles. Finally, we briefly touch on Faber series in relation to the Jump Problem following the work of H. Tietz.