On a three-dimensional analogue to the holomorphic z-powers: Laurent series expansions

Abstract

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension three using the framework of hypercomplex function theory. This second article on hand deals in particular with a generalized Laurent series expansion for the spherical shell. For this reason a new orthonormal basis of outer solid spherical monogenics in the space of square-integrable quaternion-valued functions is constructed and peculiar properties regarding the hypercomplex derivative and primitive are proved.

Keywords:

• complete orthonormal systems
• outer solid spherical monogenics
• recurrence formulae
• Laurent series expansion

AMS Subject Classifications::

• 30G35
• 32A05
• 65Q30

Acknowledgement

The author expresses his gratitude to his mentor Prof K. Gürlebeck for the helpful discussions.