Closed Forms: What They Are and Why We Care
Jonathan Michael Borwein, Richard E. Crandall · Notices of the American Mathematical Society · 2013
Closed Forms: What They AreMathematics 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". DefinitionsFirst Approach to a Definition of Closed Form.The first comes from MathWorld [56] and so may well be the first and last definition a student or other seeker-after-easy-truth finds.An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set.For example, an infinite sum would generally