Closed forms: what they are and why we care

Mathematics abounds in terms that are in frequent use yet are rarely made precise. Two such are rigorous proof and closed form (absent the technical use within differential algebra). If a rigorous proof is “that which ‘convinces’ the appropriate audience,” then a closed form is “that which looks ‘fundamental’ to the requisite consumer.” In both cases, this is a community-varying and epoch-dependent notion. What was a compelling proof in 1810 may well not be now; what is a fine closed form in 2010 may have been anathema a century ago. In this article we are intentionally informal as befits a topic that intrinsically has no one “right” answer. Let us begin by sampling the Web for various approaches to informal definitions of “closed form”.