# Schwarz differential

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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 $n= 2$.