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Zbl 0816.34040
Shackell, John
Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations. (English)
[J] Ann. Inst. Fourier 45, No.1, 183-221 (1995). ISSN 0373-0956; ISSN 1777-5310/e

Summary: 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 $x$, $e\sp x$ and $\lambda$, where $\lambda$ tends to zero at least as rapidly as some negative power of $\exp(e\sp x)$. If $\lambda$ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over ${\bbfR}(x)$.

MSC 2000:: *34E05 Asymptotic expansions (ODE)
26A12 Rate of growth of functions of one real variable
13N99 Differential algebra

Keywords: asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion

Cited in: Zbl 0835.34066

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Zentralblatt MATH Berlin [Germany]