A new integral formula for the inverse Fueter mapping theorem

TL;DR

This paper introduces a new integral formula for constructing Fueter primitives of axial monogenic functions, providing an explicit kernel description and applications to Cauchy kernels with sphere singularities.

Contribution

It presents an alternative integral method for the inverse Fueter mapping theorem, enhancing the understanding of its kernel and practical computation of Fueter primitives.

Findings

Derived a new integral formula for Fueter primitives

Explicitly described the kernel of the Fueter mapping

Applied the method to Cauchy kernels with sphere singularities

Abstract

In this paper we provide an alternative method to construct the Fueter primitive of an axial monogenic function of degree $k$, which is complementary to the one used in [F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel Journal of Mathematics, 2012]. As a byproduct, we obtain an explicit description of the kernel of the Fueter mapping. We also apply our method to obtain the Fueter primitives of the Cauchy kernels with singularities on the unit sphere.