Abstract
The theory of split-quaternion analysis and split-quaternion geometry is nowadays under full development. In this article, we extend quaternionic Hermitian Clifford analysis to the case of split-quaternions. The split-quaternionic Hermitian Dirac operator is introduced as a self-map of smooth functions defined in domains of
with values in the tensor product of the hyperbolic quaternions
, the split-quaternions
and the Clifford algebra
. It is defined by
with
being the basis of
and
denoting the twisted Hermitian Dirac operators in the split-quaternionic Clifford algebra
whose definition is based on a delicate construction of the split-quaternionic Witt basis. We establish the counterpart of the Cauchy integral formula.