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Difference between revisions of "Asymptotic sequence"

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A sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137701.png" /> such that
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A sequence of functions $\{\phi_n(x)\}$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137702.png" /></td> </tr></table>
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$$\phi_{n+1}(x)=o(\phi_n(x)),\quad x\to x_0,\quad x\in M,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137703.png" /> is a limit point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137704.png" /> (finite or infinite). If the nature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137705.png" /> is clear from the context, then one simply writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137707.png" /> is an asymptotic sequence and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137708.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137709.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377010.png" /> will also be an asymptotic sequence.
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where $x_0$ is a limit point of the set $M$ (finite or infinite). If the nature of $M$ is clear from the context, then one simply writes $x\to x_0$. If $\{\phi_n(x)\}$ is an asymptotic sequence and $\psi(x)$ is a function defined on $M$, then $\{\psi(x)\phi_n(x)\}$ will also be an asymptotic sequence.
  
 
Examples of asymptotic sequences:
 
Examples of asymptotic sequences:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377011.png" />;
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1) $\{(x-x_0)^n\},x\to x_0$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377012.png" />;
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2) $\{x^{-n}\},x\to\infty$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377013.png" />;
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3) $\{e^xx^{-n}\},x\to\infty$;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377015.png" /> is an unbounded domain in the complex plane. Asymptotic sequences such as 1), 2) and 4) are called asymptotic power sequences.
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4) $\{z^{-n}\},z\to\infty,z\in D$, where $D$ is an unbounded domain in the complex plane. Asymptotic sequences such as 1), 2) and 4) are called asymptotic power sequences.
  
  

Latest revision as of 09:48, 19 August 2014

A sequence of functions $\{\phi_n(x)\}$ such that

$$\phi_{n+1}(x)=o(\phi_n(x)),\quad x\to x_0,\quad x\in M,$$

where $x_0$ is a limit point of the set $M$ (finite or infinite). If the nature of $M$ is clear from the context, then one simply writes $x\to x_0$. If $\{\phi_n(x)\}$ is an asymptotic sequence and $\psi(x)$ is a function defined on $M$, then $\{\psi(x)\phi_n(x)\}$ will also be an asymptotic sequence.

Examples of asymptotic sequences:

1) $\{(x-x_0)^n\},x\to x_0$;

2) $\{x^{-n}\},x\to\infty$;

3) $\{e^xx^{-n}\},x\to\infty$;

4) $\{z^{-n}\},z\to\infty,z\in D$, where $D$ is an unbounded domain in the complex plane. Asymptotic sequences such as 1), 2) and 4) are called asymptotic power sequences.


Comments

References

[a1] N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[a2] A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
How to Cite This Entry:
Asymptotic sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_sequence&oldid=11934
This article was adapted from an original article by M.I. Shabunin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article