An article co-authored by Jonathan M. Borwein and the late Richard E. Crandall on closed forms has appeared in the January 2013 issue of the Notices of the American Mathematical Society. This article tries to answer the question “What is a closed form,” and then explains why obtaining a closed-form expression for a mathematical entity (as opposed, say, to a numerical value) is so important.
The full PDF of the article is available Here.
Here is the introductory paragraph of the article:
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.