We employ tools from complex analysis to construct the $$*$$ ∗ -logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $$*$$ ∗ -exponential; we establish sufficient conditions for the $$*$$ ∗ -product of two $$*$$ ∗ -exponentials to also be a $$*$$ ∗ -exponential; we calculate the slice derivative of the $$*$$ ∗ -exponential of a regular function.
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Altavilla et al. (2024) studied this question.
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