Schwarz symmetric derivative

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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:


[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)


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 , , .


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098