The \(*\)-Exponential as a Covering Map

We employ tools from complex analysis to construct the \(*\) $\ast$ -logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the \(*\) $\ast$ -exponential; we establish sufficient conditions for the \(*\) $\ast$ -product of two \(*\) $\ast$ -exponentials to also be a \(*\) $\ast$ -exponential; we calculate the slice derivative of the \(*\) $\ast$ -exponential of a regular function.