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Difference between revisions of "MacLaurin series"

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''for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621001.png" />''
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''for a function $f(z)$''
  
 
The power series
 
The power series
  
<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/m/m062/m062100/m0621002.png" /></td> </tr></table>
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$$f(z)=\sum_{k=0}^\infty\frac{f^{(k)}(0)}{k!}z^k.$$
  
It was studied by C. MacLaurin [[#References|[1]]]. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621003.png" /> analytic at zero is expanded as a [[Power series|power series]] around zero, then this series coincides with the MacLaurin series. When a function depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621004.png" /> variables, the MacLaurin series is a multiple power series:
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It was studied by C. MacLaurin [[#References|[1]]]. If a function $f(z)$ analytic at zero is expanded as a [[Power series|power series]] around zero, then this series coincides with the MacLaurin series. When a function depends on $m$ variables, the MacLaurin series is a multiple power series:
  
<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/m/m062/m062100/m0621005.png" /></td> </tr></table>
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$$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\cdots f^{(k_m)}(0)}{k_1!\cdots k_m!}z_1^{k_1}\cdots z_m^{k_m}$$
  
in which the summation is over the multi-indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621007.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062100/m0621008.png" /> are non-negative integers. A MacLaurin series is a special case of a [[Taylor series|Taylor series]].
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in which the summation is over the multi-indices $k=(k_1,\dots,k_m)$, $|k|=k_1+\dots+k_m$, and $k_i$ are non-negative integers. A MacLaurin series is a special case of a [[Taylor series|Taylor series]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. MacLaurin,  "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. MacLaurin,  "A treatise of fluxions" , '''1–2''' , Edinburgh  (1742)</TD></TR></table>

Latest revision as of 06:08, 13 June 2022

for a function $f(z)$

The power series

$$f(z)=\sum_{k=0}^\infty\frac{f^{(k)}(0)}{k!}z^k.$$

It was studied by C. MacLaurin [1]. If a function $f(z)$ analytic at zero is expanded as a power series around zero, then this series coincides with the MacLaurin series. When a function depends on $m$ variables, the MacLaurin series is a multiple power series:

$$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\cdots f^{(k_m)}(0)}{k_1!\cdots k_m!}z_1^{k_1}\cdots z_m^{k_m}$$

in which the summation is over the multi-indices $k=(k_1,\dots,k_m)$, $|k|=k_1+\dots+k_m$, and $k_i$ are non-negative integers. A MacLaurin series is a special case of a Taylor series.

References

[1] C. MacLaurin, "A treatise of fluxions" , 1–2 , Edinburgh (1742)
How to Cite This Entry:
MacLaurin series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_series&oldid=15944
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article