Àlgebra diferencial: El teorema de Liouville

Abstract

[en] Abstract It's one of the first results a maths undergraduate hears about. The function $e^{-x^{2}}$ does not have an antiderivative, the value of its integral on any given interval where it can be calculated can only be approximated via numerical methods. But what does it mean for a real function to not have an antiderivative, a function that expresses its integral in simple terms? When do we even consider a real or complex function to be expressible in simple terms? These questions are the focus of this project. Using modern results in mathematics, mainly differential algebra, we aim to introduce a theoretical frame where these questions can be posed rigorously, one where we can prove the theorem that answers them: Liouville's theorem about antiderivatives.