# Varadharaj R. Srinivasan: *Iterated and Generalized Iterated Integrals*

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. An element $x$ from a differential field extension of $k$ is called an iterated integral of (an element of) $k$ if for some positive integer $n,$ the $n-$th derivative of $x$ belongs to $k$. We show that a Picard-Vessiot extension $E$ of $k$ has a differential Galois group isomorphic to a commutative unipotent algebraic group if and only if $E$ is generated as a differential field over $k$ by an iterated integral of $k$. This result in particular refines a result of Singer and Roques (Lemma 7.5, On the algebraic dependence of holonomic functions). We introduce the notion of a generalized iterated integral of $k$ and show that a Picard-Vessiot extension $E$ of $k$ has a differential Galois group isomorphic to a unipotent algebraic group if and only if $E$ is generated as a differential field over $k$ by a generalized iterated integral of $k$. We also make some remarks on stability problems in the theory of integration in finite terms and also on the constructive inverse problem in differential Galois theory for unipotent algebraic groups.