Representation of Integral Formulas for the Extended Quaternions on Clifford Analysis

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering.