View source for Riemann derivative

''Schwarzian derivative, second symmetric derivative, of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818601.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818602.png" />''

The limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818603.png" /></td> </tr></table>

It was introduced by B. Riemann in 1854, who proved that if at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818604.png" /> the second derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818605.png" /> exists, then so does the Riemann derivative and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818606.png" />. The upper and lower limits of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818607.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818608.png" /> are called the upper (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r0818609.png" />) and lower (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081860/r08186010.png" />) Riemann derivative, respectively.

Riemann derivatives find wide application in the theory of the representation of functions by trigonometric series, and in particular in connection with the [[Riemann summation method|Riemann summation method]].



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Blaisdell  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Riemann,   "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , ''Gesammelte Math. Abhandlungen'' , Dover, reprint  (1957)  pp. 227–264</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Wolff,   "Fourier'sche Reihen" , Noordhoff  (1931)</TD></TR></table>