Difference between revisions of "Asymptotic sequence"
From Encyclopedia of Mathematics
(Importing text file) |
(details) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | A sequence of functions | + | {{TEX|done}} |
| + | 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 | + | 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) | + | 1) $\{(x-x_0)^n\},x\to x_0$; |
| − | 2) | + | 2) $\{x^{-n}\},x\to\infty$; |
| − | 3) | + | 3) $\{e^xx^{-n}\},x\to\infty$; |
| − | 4) | + | 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> | ||
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=11934
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