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 theĀ *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.