On a Cauchy-type integral related to the Helmholtz operator in the plane (Q1885510)

Vector fields and quaternionic \(\alpha\)-hyperholomorphic functions in a domain of \(\mathbb{R}^{2}\) generalizing the notion of solenoidal and irrotational vector fields are considered. For the system \[ \text{div} f=\alpha f_{0} \] \[ \text{rot} f+\alpha f=-\text{grad} f_{0} \] the sufficient conditions which the corresponding Cauchy-type integral can be continuously extendable are established. Sokhotski-Plemelj formulas are proved as well.