The Cauchy Integral Formula in Hermitian, Quaternionic and osp(4|2) Clifford Analysis

TL;DR

This paper extends the Cauchy Integral Formula to hermitian, quaternionic, and osp(4|2) Clifford analysis, revealing new integral representations for these advanced function theories in higher dimensions.

Contribution

It introduces Caychy integral formulas for osp(4|2)-monogenic functions, advancing the understanding of symmetries in quaternionic and hermitian Clifford analysis.

Findings

Established Caychy integral formulas for hermitian and quaternionic monogenic functions.

Derived new integral representations for osp(4|2)-monogenic functions.

Enhanced the theoretical framework connecting Clifford analysis and symmetry groups.

Abstract

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a corner stone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis in euclidean space R^{2n} of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on R^{2n}, quaternionic Clifford analysis arose as a further refinement by introducing a so--called hypercomplex structure Q, i.e.\ three complex structures (I, J, K) which submit to the quaternionic multiplication rules, on R^{4p}, the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Caychy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for osp(4|2)--monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.