Symbolic asymptotics: Functions of two variables, implicit functions
type: Item
instance of: scholarly article
title: Symbolic asymptotics: Functions of two variables, implicit functions
zbMATH Open document ID: 0937.41015
author: Bruno Salvy
author: John R. Shackell
published in: Journal of Symbolic Computation
publication date: 2000-06-13T00:00:00Z
review text: This paper presents an algorithmic approach to finding the limiting asymptotic behaviour of exp-log functions of two variables and hence of implicit functions of one variable. We work in \({\mathcal H}(x,y)\), the field obtained by closing \(\mathbb{K}(x,y)\) under application of exp and log, where \(\mathbb{K}\) is the field of real elementary constants. Proof that the algorithm works and terminates relies on the theory of Hardy fields. A Hardy field is a subring of the ring of germs at \(\infty\) of \(C^\infty\) functions which is a field closed under differentiation. It can be ordered by setting \(f>g\) when \(f(x)-g(x)>0\) for sufficiently large \(x\). Elements of Hardy fields tend to (possibly infinite) limits. Two elements \(f\) and \(g\) of a Hardy field tending to infinity are said to be comparable if there exists a positive integer \(n\) such that \(f<g^n\) and \(g<f^n\); thus the non-zero elements can be decomposed into comparability classes. A partial nested form of a positive function tending to zero or infinity consists essentially of a composition of exponentials and logarithms, whose length is essentially its nesting depth. A nested form is a partial nested form satisfying two extra conditions, and a nested expansion is obtained by repeatedly giving a nested form for the residual ``\(o(1)"\) part. Let \(h(x,y)\in {\mathcal H}(x,y)\) and suppose the continuous real function \(y(x)\) satisfies \(h(x,y (x))=0\). Then it is proved that \(y\) has a nested form whose length is bounded in terms of the structure of \(h\). For handling functions with finite limits, the \(z\)-function notation of [\textit{J. Shackell}, J. Symb. Comput. 10, No. 6, No. 6, 611-632 (1990; Zbl 0727.26002)], is used, where \(z\exp_n(t)\), \(z\log_n(t)\) and \(z\text{pow}_n(r,t)\) give essentially Taylor remainder terms of order \(t^{n+1}\) divided by \(t^n\), which thus have linear zeros at \(t=0\). Asymptotic estimates as \(x\to \infty\) of bivariate exp-log functions \(h(x,y(x))\) …
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 1204279
reviewed by: Francis Wright
MaRDI profile type: Publication
OpenAlex ID: W4210674847
DOI: 10.1006/JSCO.1997.0179
description: scientific article; zbMATH DE number 1204279