Volume Cauchy formulas for slice functions on real associative *-algebras

Abstract

We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For each suitable choice of a real vector subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case, we get a volume Cauchy formula. In the Clifford algebra case, the choice of the paravector subspace ℝ ⁿ⁺¹ gives a volume Cauchy formula for slice monogenic functions.

Keywords:

• Cauchy integral formula
• slice regular functions
• quaternions
• Clifford algebras

AMS Subject Classifications::

• Primary: 30G35
• Secondary: 32A30, 30E20, 13J30

Acknowledgements

The present work was partially supported by GNSAGA of INdAM.