Education

Abstract

This edition of the Education section contains two wonderfully written papers that will be very accessible to students and useful to instructors. They come from two quite different areas: complex analysis, and dynamical systems/numerical solution of differential equations. Both, in quite different ways, combine computation and mathematics. The first paper may possess one of the longer, more technical titles in the short history of the Education section, but that title serves as an excellent short abstract of the paper. The paper is "Recovering Holomorphic Functions from Their Real and Imaginary Parts without the Cauchy--Riemann Equations" by William Shaw. Written in a clear and pleasant tutorial style with exercises and examples, the paper provides a simple alternative to a fundamental component of complex analysis. It presents a purely algebraic method to recover a holomorphic (also known as analytic, or regular) complex function given just its real, or just its imaginary, part. This contrasts with the standard approach based on partial differential equations through direct application of the Cauchy--Riemann equations. The author acknowledges that the method is not entirely original, having appeared previously at least in simplified form. But apparently it is neither well known nor commonly cited, and this paper seeks to remedy that. Based upon its clarity, accessibility, and elegance, it should succeed! The paper fits any introductory course in complex analysis. It even contains a discussion and examples of a symbolic computation implementation of the approach, in Mathematica. "'The computer is always right'---wrong" and "mathematical analysis helps" could be the mantras of the second paper, "Analysis Still Matters: A Surprising Instance of Failure of Runge--Kutta--Felberg ODE Solvers." Author Joseph Skufca presents a simple dynamical system where the popular and reliable Runge--Kutta--Felberg method fails, providing an erratic solution after a certain time. Rather than revealing an interesting, perhaps chaotic solution to the system, the numerical solution simply is wrong. The culprit is a stepsize that has gotten too large and does not recover, but the real question is "why?" The paper uses simple analysis to show why the code's stepsize estimation method fails on this problem, and then shows that while typically the stepsize and solution would recover, on certain problems like this one, that is not the case. It also explains the relation between this problem and phenomena of solving stiff ordinary differential equations, and indeed, a stiff solver handles this problem well. The basic premise of the paper is that understanding of numerical simulation and the underlying mathematical analysis must go hand in hand. This is an excellent lesson that can fit in courses in basic numerical analysis, numerical differential equations, or dynamical systems, and the very clear exposition of this paper makes it ideal for teaching this lesson.