Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations

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.