Reduced quaternionic Cauchy-like integrals

Abstract The works of the Swiss mathematician Karl Rudolf Fueter and the Romanian mathematicians Grigore Constantin Moisil and Nicolae Victor Teodorescu marked the starting point of a hypercomplex analysis over the skew field of real quaternions. Nowadays, this function theory, so-called quaternionic analysis, is the most attractive and close generalization of complex analysis since it preserves many of its key features. After the discovery of the quaternions by William Rowan Hamilton occurred in 1843, the concept of reduced quaternions, referred to quaternions with vanishing third imaginary unit-coefficient, emerged. A modification of quaternionic analysis, based on functions that map from domains of three-dimensional real space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {{\mathbb{R}}^{3}} to the set of reduced quaternions, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {{\mathcal{A}}} , has recently been developed. The primary aim of the present paper is to derive an analogue of the Sokhotski–Plemelj formulas for the reduced quaternionic Cauchy-like integral in the context of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {{\mathcal{A}}} -valued functions on domains of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {{\mathbb{R}}^{3}} with Ahlfors-regular boundary, which is considered distinct from that of quaternionic analysis, despite their natural similarities. Our second task is to establish a Poincaré–Bertrand formula for interchanging the order of integration of two repeated singular reduced quaternionic Cauchy-like integrals, drawing on arguments involving quaternionic analysis. Finally, a Geza Freud-type theorem is proved, as well as some local properties of the singular reduced quaternionic Cauchy-like integral.