  Carlini's career was mainly dedicated to astronomy, but he was also a
particularly skilled mathematician. In this article we collect and analyse his
mathematical contributions in detail. In particular, in his important Memoir of
the year 1817 devoted to Kepler's equation he introduced an innovative idea to
solve ordinary differential equations with singular perturbations by means of
asymptotic expansions. In the same Memoir also appeared, five years before
Laplace's contributions, what is usually called the Laplace limit constant.
Furthermore, Carlini published other mathematical Memoirs anticipating, 70
years in advance, the importance of complex branches of the Lambert's special
function.
