Working with Families of Inverse Functions

Abstract

When evaluating or simplifying mathematical expressions, the question arises of how to handle inverse functions. The problem is that for a non-injective function $f$ $: D \rightarrow R$, the inverse is generally not a function $R \rightarrow D$ since there may be multiple pre-images for a given point. The majority of work in this area has fallen into two camps: either the inverse functions, and expressions involving them, are treated as multi-valued objects, or inverse functions are taken to have one principal value. Both these approaches lead to difficulties in evaluation and simplification. It is possible to define the inverse as a function from R to sets of elements of D, but then the algebra of expressions involving the inverse becomes overly complicated. This article extends previous work based on a different approach: instead, the inverse of a function is taken to be a labelled family of functions, with the label specifying the pre-image in the original function’s domain. This convention is already used by some authors for logarithms, but it can be applied more generally. In some cases, the branch indices can appear in identities that give more broadly applicable simplification rules. In this paper we survey how this approach can be applied to elementary functions, including the Lambert W.