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. Sommario La carriera di Carlini fu prevalentemente dedicata all'astronomia, ma egli fu anche un matematico particolarmente abile. In questo articolo raccogliamo e analizziamo in dettaglio i suoi contributi matematici. In particolare, nella sua importante Memoria dell'anno 1817 dedicata all'equazione di Keplero, egli introdusse un'idea innovativa per risolvere, mediante sviluppi asintotici, equazioni differenziali ordinarie con perturbazioni singolari. Nella stessa Memoria apparve anche, cinque anni prima dei contributi dati da Laplace, quella che viene solitamente chiamata la costante di Laplace. Inoltre, Carlini pubblicò altre memorie matematiche considerando, con 70 anni di anticipo, i rami della funzione speciale di Lambert nel piano complesso.
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