View source for Schwarz symmetric derivative

''of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835801.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835802.png" />''

The value

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It is sometimes called the [[Riemann derivative|Riemann derivative]] or the second symmetric derivative. For the first time introduced by B. Riemann in 1854 (see [[#References|[2]]]); it was studied by H.A. Schwarz [[#References|[1]]]. More generally, the symmetric derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835805.png" /> is also called a Schwarz symmetric derivative:

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.A. Schwarz,   "Beweis eines für die Theorie der trigonometrischen Reihen in Betracht kommenden Hülfssatzes" , ''Gesammelte Math. Abhandlungen'' , Chelsea, reprint  (1972)  pp. 341–343</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835808.png" /> (1868)))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>



====Comments====
The name general derivative is also used for this notion. A natural approach is to start with the central difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s0835809.png" />, and to define the first symmetric derivative as

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and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s08358011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s08358012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083580/s08358013.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>