Namespaces
Variants
Actions

Schwarz differential

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


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=55234
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article