The inverse Fueter mapping theorem in integral form using spherical monogenics (Q1955751)

The Fueter mapping theorem states that given a holomorphic function \(f\) of a paravector variable defined on an axially symmetric open set \(U \subset\mathbb R^{n+1}\), where \(n\) is a odd number, then, for \(\mathcal P_k(\underline{x})\) a homogeneous monogenic polynomial of degree \(k\) and \(\Delta_x\) the Laplace operator in dimension \(n+1\), the function \( \breve{f}(x)\mathcal P_k(\underline{x})\) given by \[ \Delta_x^{\frac{n-1}{2}}\left( f(x)\mathcal P_k(\underline{x}) \right) = \breve{f}(x)\mathcal P_k(\underline{x}) \] is a monogenic function on \(U\). In this paper an integral representation formula of \(f(x)\mathcal P_k(\underline{x})\) is determined in terms of \(\breve{f}(x)\mathcal P_k(\underline{x}),\) allowing a Fueter primitive to be obtained for any monogenic function on an axially symmetric open set.