On boundary values for rectifiable curves of a generalization of the Cauchy-type integral related to the Helmholtz operator in $R^2$

TL;DR

This paper investigates boundary value conditions for generalized Cauchy-type integrals related to the Helmholtz operator in two dimensions, extending classical results to quaternionic hyperholomorphic functions and solenoidal/irrotational fields.

Contribution

It establishes sufficient conditions for the continuous extension of these integrals and proves Sokhotski-Plemelj-type formulas in this generalized setting.

Findings

Sufficient boundary conditions for integral extension are identified.

Sokhotski-Plemelj formulas are generalized for quaternionic hyperholomorphic functions.

Results connect boundary behavior with Helmholtz-related vector fields.

Abstract

There are considered vector fields and quaternionic $\alpha$-hyperholomorphic functions in a domain of $R^2$ which generalize the notion of solenoidal and irrotational vector fields. There are established sufficient conditions for the corresponding Cauchy-type integral along a closed Jordan rectifiable curve to be continuously extended onto the closure of a domain. The Sokhotski-Plemelj-type formulas are proved as well.