In this paper we show that for each formal symbol $p$ of minimum type there is a formal symbol $q$ of minimum type satisfying $\exp (:p:) = : \exp (q):$.
"The exponential calculus of pseudodifferential operators of minimum type. I." Proc. Japan Acad. Ser. A Math. Sci. 89 (1) 6 - 10, January 2013. https://doi.org/10.3792/pjaa.89.6