$\ast$-logarithm for slice regular functions

Abstract

In this paper, we study the (possible) solutions of the equation ${\exp }_{\ast }(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $\mathbb{H}$ and ${\exp }_{\ast }$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies ${\exp }_{\ast }(f)=g$ is called a $\ast$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part $g_v$ of $g$, for the existence of a $\ast$-logarithm of $g$, under a natural topological condition on the domain $\Omega$. By this way, we prove an existence result if $g_v$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_v$ are finite, the domain is either the unit ball, or $\mathbb{H}$, or $\mathbb{D}$ (the solid torus obtained by circularization in $\mathbb{H}$ of the disc contained in $\mathbb{C}$ and centered in $2\sqrt{-1}$ with radius $1$), and a further condition on the “real part” $g_0$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_v$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.