On a three dimensional analogue to the holomorphic z-powers

The main objective of this article is a constructive generalization of the holomorphic power and Laurent series expansions in C to dimension 3 using the framework of hypercomplex function theory. For this reason, deals the first part of this article with generalized Fourier & Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in R^3 and their series coefficients can be explicitly (one-to-one) linked with each other. Furthermore, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented. The latter results are then used to construct a new orthonormal bases of outer solid spherical monogenics in the space of square integrable quaternion-valued functions. This finally leads to the definition of a generalized Laurent series expansion for the spherical shell.

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