Namespaces
Variants
Actions

Difference between revisions of "Riemann derivative"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''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" />''
+
<!--
 +
r0818601.png
 +
$#A+1 = 10 n = 0
 +
$#C+1 = 10 : ~/encyclopedia/old_files/data/R081/R.0801860 Riemann derivative,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
''Schwarzian derivative, second symmetric derivative, of a function  $  f $
 +
at a point $  x _ {0} $''
  
 
The limit
 
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>
+
$$
 +
D  ^ {2} f( x _ {0} )  = \
 +
\lim\limits _ {h \rightarrow 0 } 
 +
\frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} - h) }{h  ^ {2}
 +
}
 +
.
 +
$$
  
It 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
+
It was introduced by B. Riemann in 1854, who proved that if at a point $  x _ {0} $
 +
the second derivative $  f ^ { \prime\prime } ( x _ {0} ) $
 +
exists, then so does the Riemann derivative and $  D  ^ {2} f( x _ {0} ) = f ^ { \prime\prime } ( x _ {0} ) $.  
 +
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.
+
\frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} + h) }{h  ^ {2} }
  
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]].
+
$$
  
 +
as  $  h \rightarrow 0 $
 +
are called the upper ( $  {\overline{D}\; } {}  ^ {2} f( x _ {0} ) $)
 +
and lower ( $  \underline{D}  ^ {2} f( x _ {0} ) $)
 +
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]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:11, 6 June 2020


Schwarzian derivative, second symmetric derivative, of a function $ f $ at a point $ x _ {0} $

The limit

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

It was introduced by B. Riemann in 1854, who proved that if at a point $ x _ {0} $ the second derivative $ f ^ { \prime\prime } ( x _ {0} ) $ exists, then so does the Riemann derivative and $ D ^ {2} f( x _ {0} ) = f ^ { \prime\prime } ( x _ {0} ) $. The upper and lower limits of

$$ \frac{f( x _ {0} + h) - 2f( x _ {0} ) + f( x _ {0} + h) }{h ^ {2} } $$

as $ h \rightarrow 0 $ are called the upper ( $ {\overline{D}\; } {} ^ {2} f( x _ {0} ) $) and lower ( $ \underline{D} ^ {2} f( x _ {0} ) $) 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.

Comments

References

[a1] T.M. Apostol, "Mathematical analysis" , Blaisdell (1957)
[a2] B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264
[a3] J. Wolff, "Fourier'sche Reihen" , Noordhoff (1931)
How to Cite This Entry:
Riemann derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_derivative&oldid=48544
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article