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Difference between revisions of "Riemann summation method"

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A method for summing series of numbers. A series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820301.png" /> can be summed by Riemann's method to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820302.png" /> if
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A method for summing series of numbers. A series $\sum_{n=0}^\infty a_n$ can be summed by Riemann's method to a number $S$ if
  
<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/r082/r082030/r0820303.png" /></td> </tr></table>
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$$\lim_{h\to0}\left[a_0+\sum_{n=1}^\infty a_n\left(\frac{\sin nh}{nh}\right)^2\right]=S.$$
  
 
This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see [[#References|[1]]]). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A [[Trigonometric series|trigonometric series]]
 
This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see [[#References|[1]]]). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A [[Trigonometric series|trigonometric series]]
  
<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/r082/r082030/r0820304.png" /></td> </tr></table>
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$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$
  
with bounded coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820305.png" /> can be summed by Riemann's method at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820306.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820307.png" /> if the function
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with bounded coefficients $a_n,b_n$ can be summed by Riemann's method at a point $x_0$ to a number $S$ if the function
  
<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/r082/r082030/r0820308.png" /></td> </tr></table>
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$$F(x)=\frac{a_0x^2}{4}-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}$$
  
has, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820309.png" />, [[Riemann derivative|Riemann derivative]] equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r08203010.png" />.
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has, at $x_0$, [[Riemann derivative|Riemann derivative]] equal to $S$.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Regularity (cf. [[Regular summation methods|Regular summation methods]]) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r08203011.png" /> is also called the [[Riemann function|Riemann function]].
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Regularity (cf. [[Regular summation methods|Regular summation methods]]) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function $F(x)$ is also called the [[Riemann function|Riemann function]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekman,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekman,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>

Latest revision as of 16:17, 4 October 2014

A method for summing series of numbers. A series $\sum_{n=0}^\infty a_n$ can be summed by Riemann's method to a number $S$ if

$$\lim_{h\to0}\left[a_0+\sum_{n=1}^\infty a_n\left(\frac{\sin nh}{nh}\right)^2\right]=S.$$

This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see [1]). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A trigonometric series

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$

with bounded coefficients $a_n,b_n$ can be summed by Riemann's method at a point $x_0$ to a number $S$ if the function

$$F(x)=\frac{a_0x^2}{4}-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}$$

has, at $x_0$, Riemann derivative equal to $S$.

References

[1] B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[3] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[4] G.H. Hardy, "Divergent series" , Clarendon Press (1949)


Comments

Regularity (cf. Regular summation methods) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function $F(x)$ is also called the Riemann function.

References

[a1] W. Beekman, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Riemann summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_summation_method&oldid=33493
This article was adapted from an original article by T.P. Lykashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article