Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

John Shackell

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 1, page 183-221
  • ISSN: 0373-0956

Abstract

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We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.Let g be an element of a Hardy field which has an asymptotic series expansion in x , e x and λ , where λ tends to zero at least as rapidly as some negative power of exp ( e x ) . If λ actually occurs in the expansion, then g cannot satisfy a first-order algebraic differential equation over ℝ ( x ) .

How to cite

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Shackell, John. "Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations." Annales de l'institut Fourier 45.1 (1995): 183-221. <http://eudml.org/doc/75113>.

@article{Shackell1995,
abstract = {We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.Let $g$ be an element of a Hardy field which has an asymptotic series expansion in $x$, $e^x$ and $\lambda $, where $\lambda $ tends to zero at least as rapidly as some negative power of $\exp (e^x)$. If $\lambda $ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over $\{\Bbb R\}(x)$.},
author = {Shackell, John},
journal = {Annales de l'institut Fourier},
keywords = {asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion},
language = {eng},
number = {1},
pages = {183-221},
publisher = {Association des Annales de l'Institut Fourier},
title = {Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations},
url = {http://eudml.org/doc/75113},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Shackell, John
TI - Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 1
SP - 183
EP - 221
AB - We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.Let $g$ be an element of a Hardy field which has an asymptotic series expansion in $x$, $e^x$ and $\lambda $, where $\lambda $ tends to zero at least as rapidly as some negative power of $\exp (e^x)$. If $\lambda $ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over ${\Bbb R}(x)$.
LA - eng
KW - asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion
UR - http://eudml.org/doc/75113
ER -

References

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