# Schwarz symmetric derivative

From Encyclopedia of Mathematics

*of a function at a point *

The value

It is sometimes called the Riemann derivative or the second symmetric derivative. For the first time introduced by B. Riemann in 1854 (see [2]); it was studied by H.A. Schwarz [1]. More generally, the symmetric derivative of order is also called a Schwarz symmetric derivative:

#### References

[1] | H.A. Schwarz, "Beweis eines für die Theorie der trigonometrischen Reihen in Betracht kommenden Hülfssatzes" , Gesammelte Math. Abhandlungen , Chelsea, reprint (1972) pp. 341–343 |

[2] | B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , B. Riemann's Gesammelte Mathematische Werke , Dover, reprint (1953) pp. 227–271 ((Original: Göttinger Akad. Abh. (1868))) |

[3] | I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) |

[4] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |

[5] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

#### Comments

The name general derivative is also used for this notion. A natural approach is to start with the central difference , and to define the first symmetric derivative as

and then , , .

#### References

[a1] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 |

**How to Cite This Entry:**

Schwarz symmetric derivative. T.P. Lukashenko (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Schwarz_symmetric_derivative&oldid=15458

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098