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

Annales de l'institut Fourier (1995)

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

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topShackell, 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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