Orders of Infinity

Many problems in analysis (as well as adjacent fields such as combinatorics, theoretical computer science, and PDE) are interested in the order of growth (or decay) of some quantity that depends on one or more asymptotic parameters (such as ) – for instance, whether the quantity grows or decays linearly, quadratically, polynomially, exponentially, etc. in . In the case where these quantities grow to infinity, these growth rates had once been termed “orders of infinity” – for instance, in the 1910 book of this name by Hardy – although this term has fallen out of use in recent years. (Hardy fields are still a thing, though.) In modern analysis, asymptotic notation is the preferred device to organize orders of infinity. There are a couple of flavors of this notation, but here is one such (a blend of Hardy’s notation and Landau’s notation). Formally, we need a parameter space equipped with a non-principal filter that describes the subsets of parameter space that are “sufficiently large” ( ... Read full article.