Education

Abstract

A long time ago, at an institution, far, far away, I was fortunate enough to study asymptotics with one of the masters of the subject (Steve Orszag). I was rapidly seduced by what has been called the "Devil's Invention"---asymptotic series, which, while divergent, are enormously useful in both analysis and computation. While mathematicians and engineers often learn how to use asymptotic methods, the mathematical underpinnings of these remarkably powerful tools are often lost in introductory courses. This issue's Education section has a contribution from another master of the subject; John P. Boyd has written a captivating article, "Hyperasymptotics and the Linear Boundary Layer Problem: Why Asymptotic Series Diverge," which uses a simple problem to explore why asymptotic series usually diverge and how to optimally truncate these series and bound their error. The article is perfectly pitched for an apprentice's first sorties into the subject. Boyd uses a simple boundary layer problem to illustrate a menagerie of possible behaviors for asymptotic series and helps the reader navigate the treacherous and variegated shores of asymptopia. This article also offers a perspective on the modern development of the subject; the quest for better approximation has led to the development of hyperasymptotics in the past few decades. This article will make a marvelous companion to any course on asymptotic methods. It should help alleviate a student's mistrust of asymptotic series, elevating the "Devil's Invention" to a trusted weapon in the aspiring mathematician's arsenal.