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.

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Francesco Carlini：开普勒方程和奇异微分方程的渐近解

卡里尼的职业生涯主要致力于天文学，但他也是一位特别熟练的数学家。在本文中，我们将详细收集和分析他的数学贡献。特别是，在他 1817 年专门讨论开普勒方程的重要回忆录中，他引入了一个创新的想法，通过渐近展开来求解具有奇异摄动的常微分方程。在同一本回忆录中也出现了，在拉普拉斯贡献的五年前，通常被称为拉普拉斯极限常数。此外，卡里尼出版了其他数学回忆录，提前 70 年预测了兰伯特特殊函数的复杂分支的重要性。