  The Alekseev-Meinrenken diffeomorphism is a distinguished diffeomorphism from
the space of $n\times n$ Hermitian matrices to the space of $n\times n$
positive definite Hermitian matrices. This paper derives the explicit
expression of the diffeomorphism, via the Stokes phenomenon of meromorphic
linear systems of ordinary differential equations with Poncar\'{e} rank $1$.
