2 - Singularities in Action

Summary

Singularities are central to treating the boundary eigenvalue problems in this book, both singularities of differential equations and those of their solutions. Poincaré was probably the first to recognise their importance and treat them conceptually, by introducing what he called the rank. However, I have chosen a slightly different definition, introducing the ’singularity’ s-rank. With this definition, the non-elementary regular singularity is standard, with s-rank 1. Given this concept, the singularities of our treated differential equations always have half-integer s-rank, because of the order (2) of the underlying differential equation. Moreover, regular and irregular singularities are distinguished, for s-rank larger than 1 or not. There are two types of regular singularities – s-rank 1 and s-rank 1/2 – the latter called elementary singularities. Among the irregular singularities are those having integer s-rank and odd half-integer s-rank. The irregular singularity whose s-rank is smallest is R = 3/2. The standard singularity is not – as with Poincaré – the elementary one, but the non-elementary regular singularity of the underlying differential equation with s-rank 1.