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Difference between revisions of "Schwarz differential"

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The principal part of the [[Schwarz symmetric derivative|Schwarz symmetric derivative]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083500/s0835001.png" />. More precisely, if for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083500/s0835002.png" /> of a real variable,
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<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/s/s083/s083500/s0835004.png" /></td> </tr></table>
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The principal part of the [[Schwarz symmetric derivative|Schwarz symmetric derivative]] of order  $  n $.  
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More precisely, if for a function  $  f $
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of a real variable,
  
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then the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083500/s0835005.png" /> is called the Schwarz differential of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083500/s0835006.png" />. When a Schwarz differential is mentioned without specifying the order, it is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083500/s0835007.png" />.
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$$
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\Delta  ^ {n} f ( x, \Delta x)  = \sum _ { k= } 0 ^ { n }  \left (
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\begin{array}{c}
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n \\
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k
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\end{array}
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\right ) (- 1)  ^ {k}
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f \left ( x + n-
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\frac{2k}{2}
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\Delta x \right ) =
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$$
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$$
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= \
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A \cdot ( \Delta x)  ^ {n} + o(( \Delta x)  ^ {n} ),
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$$
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then the expression  $  A \cdot ( \Delta x)  ^ {n} $
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is called the Schwarz differential of order $  n $.  
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When a Schwarz differential is mentioned without specifying the order, it is usually assumed that $  n= 2 $.

Revision as of 08:12, 6 June 2020


The principal part of the Schwarz symmetric derivative of order $ n $. More precisely, if for a function $ f $ of a real variable,

$$ \Delta ^ {n} f ( x, \Delta x) = \sum _ { k= } 0 ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f \left ( x + n- \frac{2k}{2} \Delta x \right ) = $$

$$ = \ A \cdot ( \Delta x) ^ {n} + o(( \Delta x) ^ {n} ), $$

then the expression $ A \cdot ( \Delta x) ^ {n} $ is called the Schwarz differential of order $ n $. When a Schwarz differential is mentioned without specifying the order, it is usually assumed that $ n= 2 $.

How to Cite This Entry:
Schwarz differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_differential&oldid=48629
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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