In 1893, Felix Klein (of Klein bottle fame) said
It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.
Gerhard Kristensson’s book Second Order Differential Equations: Special Functions and Their Classification concerns precisely what Felix Klein called “the central problem of the whole of modern [i.e. late nineteenth century] mathematics.” What was once considered central is now considered arcane. The special functions arising from differential equations in mathematical physics remain extraordinarily useful, though they are less familiar these days. Special functions fall between two stools, too advanced for a typical undergraduate curriculum and yet not a popular area of research and hence not part of the graduate curriculum.
As Kristensson points out, there are three popular approaches to classifying and presenting special functions. These methods use as their organizing principles second order differential equations, Lie groups, and integral averages respectively. The Lie group and integral average approaches are more modern. Kristensson takes the more traditional approach: classification based on differential equations, particularly the singular behavior of the coefficients.
Sophomore differential equation classes touch on special functions when they present power series solution techniques. However, such classes barely scratch the surface. When I first saw this material I had the impression that solutions near a singular point were a messy footnote to the general theory and hence hardly covered by our text. However, the solutions near singular points are the heart of the theory from the perspective of mathematical physics and special functions.
Series solutions of differential equations with singular coefficients are not often thoroughly studied in undergraduate courses for good reasons. The material is complex, both in the colloquial sense of complexity and in the mathematical sense of requiring analytic function theory.
The development of Second Order Differential Equations is tedious. The book is well-written, but the subject matter is intricate. A big picture of special function relationships emerges by the end and the book has several helpful diagrams to help visualize these relationships. Along the way, however, a reader could easily get lost in details. An instructor teaching from this book might do well to have students flip ahead to these diagrams early in the course as a map to the final destination.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs at The Endeavour.