On the generalized Cauchy transform of power functions

Abstract.

In this paper the bi-axial splitting $\mathbb{R}^{m} = \mathbb{R}^{m_1} \oplus \mathbb{R}^{m_2}$ is considered and a Cauchy kernel is obtained by applying the inversion operator in $\mathbb{R}^{m}$ on a monogenic function on $\mathbb{R}^{m_1}.$ This gives rise to a generalized Cauchy transformation and by means of its action on inner and outer power functions, monogenic generalized power functions are obtained.