Closed Forms by Borwein and Crandall Essay Example | Topics and Well Written Essays - 2000 words

Closed Forms by Borwein and Crandall - Essay Example Third section deals with detailed examples on closed forms. Next, recent examples of advanced research on closed forms are discussed. Then there is the fifth section titled â€Å"profound curiosities† (Borwein and Crandall 2010, p. 24) followed by the concluding section of the article. In the concluding section, several open questions have been discussed. The first section of this paper is particularly important because it explains the very significance of this article. In this section, the authors attempt to furnish a definition of closed form. But in doing this, the authors revisit a basic concept of mathematics, that is the concept of rigorous proof. The authors wish to furnish a rigorous definition of closed forms with the help of the concept of rigorous proof. However, the problem is that the general notion of rigorous proof is a kind of â€Å"community-varying and epoch dependent† concept (Borwein and Crandall 2010, p. 1). Consequently, even a potential rigorous d efinition of closed forms is likely to provide an exhaustive treatment to the matter. 2. Discussion The authors have adopted seven different approaches to define a closed form. The first three approaches are very basic and theoretical in nature. The fourth approach chiefly utilises set algebra with particular focus on exponential and logarithmic functions. Using this approach, Chow (1999) remarks that the term closed form must imply explicit in the sense that the expression in closed form is meaningful, clearly open to all calculations and standard mathematical operators can be applied (Borwein and Crandall 2010, section 1.0.4). Although most algebraic functions do not have a simple explicit expression, scientists and mathematicians are trying to introduce concepts like hyperclosure and superclosure. The fifth approach is again elementary in nature with emphasis on theory rather than correlative analysis with respect to sufficiently complicated equations and identities (Borwein and Crandall 2010). In discussing the sixth approach, the authors have put their own input to refine the understanding of this concept as deduced from previous research works of experts like Bailey, Borwein, and Crandall (2008). First, the Borwein and Crandall (2010) consider any convergent sum given by the following expression: x = ?cnzn (where x is a member of the set X) †¦ †¦ †¦ (1) Explaining the different variables and operators that are seen in (1), we must mention that c0 is rational; z is algebraic; and n ? 0. Furthermore, for n > 0 we have: , where B and A are integer polynomials such that deg B ? deg A. Also, the set X contains generalised hypergeometric evaluations as established by the authors (Borwein and Crandall 2010, section 1.2.2) as a part of the ring of hyperclosure denoted by H (which is begot from all generalised hypergeometric evaluations). Now according to the authors: â€Å"Under these conditions the expansion for x converges absolutely on the ope n disk |z| < 1. However, we also allow x to be any finite analytic-continuation value of such a series; moreover, when z lies on a branch cut we presume both branch limits to be elements of X. (See ensuing examples for some clarification.) It is important to note that our set X is closed under rational multiplication, due to freedom of choice for c0. † (Borwein and Crandall 2010, section 1.2.2) The merit of this approach is that it introduces us to the concept of hyperclosure.