# Schwarz symmetric derivative

of a function $f$ at a point $x _ {0}$

The value

$$D ^ {2} f( x _ {0} ) = \lim\limits _ {h \rightarrow 0 } \frac{f( x _ {0} + h)- 2f( x _ {0} )+ f( x _ {0} - h) }{h ^ {2} } .$$

It is sometimes called the Riemann derivative or the second symmetric derivative. For the first time introduced by B. Riemann in 1854 (see ); it was studied by H.A. Schwarz . More generally, the symmetric derivative of order $n$ is also called a Schwarz symmetric derivative:

$$D ^ {n} f( x) = \lim\limits _ {h \rightarrow 0 } \frac{\Delta _ {h} ^ {n} f( x) }{h ^ {n} } =$$

$$= \ \lim\limits _ {h \rightarrow 0 } \frac{\sum _ { k= } 0 ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f ( x + ( n- 2k ) h / 2 ) }{h ^ {n} } .$$

How to Cite This Entry:
Schwarz symmetric derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_symmetric_derivative&oldid=48635
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article