Difference between revisions of "MacLaurin series"
From Encyclopedia of Mathematics
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| − | ''for a function | + | {{TEX|done}} |
| + | ''for a function $f(z)$'' | ||
The power series | The power series | ||
| − | + | $$f(z)=\sum_{k=0}^\infty\frac{f^{(k)}(0)}{k!}z^k.$$ | |
| − | It was studied by C. MacLaurin [[#References|[1]]]. If a function | + | 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: |
| − | + | $$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\dots f^{(k_m)}(0)}{k_1!\dots k_m!}z_1^{k_1}\dots z_m^{k_m}$$ | |
| − | in which the summation is over the multi-indices | + | 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> | ||
Revision as of 19:27, 21 November 2018
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:
$$\sum_{|k|=0}^\infty\frac{f^{(k_1)}(0)\dots f^{(k_m)}(0)}{k_1!\dots k_m!}z_1^{k_1}\dots z_m^{k_m}$$
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
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