$*$-Logarithm for Slice Regular Functions

In this paper, we study the (possible) solutions of the equation $\exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $\mathbb{H}$ and $\exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $\exp_{*}(f)=g$ is called a $*$-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 $*$-logarithm of $g$, under a natural topological condition on the domain $\Omega$. By the 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 $\mathbb{D}$ 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.