View source for Asymptotic sequence

A sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a0137701.png" /> 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>

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.

Examples of asymptotic sequences:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377011.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377012.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013770/a01377013.png" />;

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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====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>