Cauchy an Innovative Mathematician: the Fundamentals of Infinitesimal Calculus, the Theorem of Finite Increments and the Functions of an "Imaginary" Variable

Calculus is the founding branch of mathematical analysis that studies the "local behavior" of a function through the notions of continuity and limit, used in almost all fields of mathematics and physics and science in general. In the article we wanted to highlight through Cauchy the objectives of infinitesimal analysis expand and include complex analysis. In the second part of the work we concentrated on the finite increment theorem. It is one of the classical theorems of Mathematical Analysis, whose importance is justified by the fact that Lagrange's theorem turns out to be a trivial consequence. Finally, we will highlight its contribution to the functions of an imaginary variable: it is a contribution of fundamental importance for mathematics scholars of all times.