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Difference between revisions of "Asymptotic sequence"

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3) $\{e^xx^{-n}\},x\to\infty$;
 
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.
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4) $\{z^{-n}\},z\to\infty,z\in D$, where $D$ is an unbounded domain in the complex plane.
 
 
 
 
 
 
====Comments====
 
  
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Asymptotic sequences such as 1), 2) and 4) are called asymptotic power sequences.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.G. de Bruijn,   "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Erdélyi,   "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Erdélyi, "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR>
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</table>

Latest revision as of 19:31, 13 April 2024

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.

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=55714
This article was adapted from an original article by M.I. Shabunin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article