Hardy fields and existence of transexponential functions
type: Item
instance of: scholarly article
title: Hardy fields and existence of transexponential functions
zbMATH Open document ID: 0593.26003
DOI: 10.1007/BF02189932
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137133
review text: The main result of this paper is that every Hardy field, which is maximal [cf. \textit{N. Boubaki}: Eléments de mathématique. Fasc. XII. Première partie. Livre IV: Fonctions d'une variable réelle (théorie élémentaire) (1961; Zbl 0131.050), p. 107] under \(\gg\) (greater for sufficiently large values of the variable), is unbounded. It is also shown that there exist transexponential functions \((f\gg e_ k\) for all positive k, where \(e_ k\) is the k-th iterate of the exponential function) in Hardy fields and that there exist (transexponential) analytic solutions f with \(\lim f=\infty,\) \(f\gg 0,\) of the functional equation \(f(x+1)=e^{f(x)}\) and also of \(f(x+1)=e^{f(x)}-1,\) such that solutions of the former equation satisfy \(f(x)f'(x-1)=f'(x)\) (the statements in the paper are stronger). Finally properties of rational iterates of the exponential function are examined and four unsolved problems are listed (a note added in proof states that the author has since solved the first two).
Mathematics Subject Classification ID: 26A12
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 3953127
zbMATH Keywords: Abel's functional equation
zbMATH Keywords: algebraic functional differential
zbMATH Keywords: equations
zbMATH Keywords: transexponential functions
zbMATH Keywords: Hardy fields
zbMATH Keywords: rational iterates of the exponential function
author: Michael Boshernitzan
MaRDI profile type: Publication
cites work: Q3919976
cites work: Some results on hypertranscendental meromorphic functions
cites work: An extension of Hardy's class L of ''orders of infinity''
cites work: New ''orders of infinity''
cites work: "Orders of Infinity" Generated by Difference Equations
cites work: Discrete "Orders of Infinity"
cites work: Q5339415
cites work: Analytic Hardy Fields and Exponential Curves in the Real Plane
cites work: Q5592596
cites work: Differential valuations
cites work: Hardy fields
cites work: The Rank of a Hardy Field
cites work: Rank change on adjoining real powers to Hardy fields
cites work: Fractional iteration of exponentially growing functions
cites work: Regular iteration of real and complex functions
description: scientific article; zbMATH DE number 3953127