Difference between revisions of "MacLaurin formula"
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| − | + | A particular case of the [[Taylor formula|Taylor formula]]. Let a function $ f $ | |
| + | have $ n $ | ||
| + | derivatives at $ x = 0 $. | ||
| + | Then in some neighbourhood $ U $ | ||
| + | of this point $ f $ | ||
| + | can be represented in the form | ||
| − | + | $$ | |
| + | f ( x) = \ | ||
| + | \sum _ { k= } 0 ^ { n } | ||
| + | \frac{f ^ { ( k) } ( 0) }{k ! } | ||
| + | x ^ {k} + r _ {n} ( x) ,\ \ | ||
| + | x \in U , | ||
| + | $$ | ||
| + | where $ r _ {n} ( x) $, | ||
| + | the $ n $- | ||
| + | th order remainder term, can be represented in some form or other. | ||
| + | |||
| + | The term "MacLaurin formula" is also used for functions of $ m $ | ||
| + | variables $ x = ( x _ {1} \dots x _ {m} ) $. | ||
| + | In this case $ k $ | ||
| + | in the MacLaurin formula is taken to be a multi-index, $ k = ( k _ {1} \dots k _ {m} ) $( | ||
| + | see [[MacLaurin series|MacLaurin series]]). The formula is named after C. MacLaurin. | ||
====Comments==== | ====Comments==== | ||
| − | For some expressions for the remainder | + | For some expressions for the remainder $ r _ {n} ( x) $ |
| + | and for estimates of it see [[Taylor formula|Taylor formula]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD></TR></table> | ||
Revision as of 04:11, 6 June 2020
A particular case of the Taylor formula. Let a function $ f $
have $ n $
derivatives at $ x = 0 $.
Then in some neighbourhood $ U $
of this point $ f $
can be represented in the form
$$ f ( x) = \ \sum _ { k= } 0 ^ { n } \frac{f ^ { ( k) } ( 0) }{k ! } x ^ {k} + r _ {n} ( x) ,\ \ x \in U , $$
where $ r _ {n} ( x) $, the $ n $- th order remainder term, can be represented in some form or other.
The term "MacLaurin formula" is also used for functions of $ m $ variables $ x = ( x _ {1} \dots x _ {m} ) $. In this case $ k $ in the MacLaurin formula is taken to be a multi-index, $ k = ( k _ {1} \dots k _ {m} ) $( see MacLaurin series). The formula is named after C. MacLaurin.
Comments
For some expressions for the remainder $ r _ {n} ( x) $ and for estimates of it see Taylor formula.
References
| [a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108 |
How to Cite This Entry:
MacLaurin formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_formula&oldid=47742
MacLaurin formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_formula&oldid=47742
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article