View source for Asymptotic power series

An asymptotic series with respect to the sequence

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or with respect to a sequence

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(cf. [[Asymptotic expansion|Asymptotic expansion]] of a function). Asymptotic power series may be added, multiplied, divided and integrated just like convergent power series.

Let two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137504.png" /> have the following asymptotic expansions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137505.png" />:

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Then

1)

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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a0137508.png" /> are constants);

2)

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3)

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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375011.png" /> are calculated as for convergent power series);

4) if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375012.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375013.png" />, then

<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/a013750/a01375014.png" /></td> </tr></table>

5) an asymptotic power series cannot always be differentiated, but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375015.png" /> has a continuous derivative which can be expanded into an asymptotic power series, then

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Examples of asymptotic power series.

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375019.png" /> is the Hankel function of order zero (cf. [[Hankel functions|Hankel functions]]) (the above asymptotic power series diverge for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375020.png" />).

Similar assertions are also valid for functions of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375022.png" /> in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375023.png" /> is regular in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375024.png" /> and if

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uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375026.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375027.png" /> inside any closed angle contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375028.png" />, then

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uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375030.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013750/a01375031.png" /> in any closed angle contained in D.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Copson,   "Asymptotic expansions" , Cambridge Univ. Press  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Erdélyi,   "Asymptotic expansions" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. de Bruijn,   "Asymptotic methods in analysis" , Dover, reprint  (1981)</TD></TR></table>