Difference between revisions of "Riemann function"
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(Created page with "The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$ =Properties=...") |
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The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by | The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by | ||
$$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$ | $$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$ | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Riemannplot.png|Plot of $R(x)$ on $[0,1]$. | ||
| + | File:Riemannfunction.gif|The partial sum $R(x,N)=\displaystyle\sum_{k=1}^N \dfrac{\sin(k^2 x)}{k^2}$ for various values of $N$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
| − | < | + | [[Riemann function is continuous]]<br /> |
| − | + | [[Riemann function is almost nowhere differentiable]]<br /> | |
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| − | + | =References= | |
| − | + | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br /> | |
| − | </ | ||
| − | + | [[Category:SpecialFunction]] | |
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Latest revision as of 03:26, 6 July 2016
The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$
Plot of $R(x)$ on $[0,1]$.
The partial sum $R(x,N)=\displaystyle\sum_{k=1}^N \dfrac{\sin(k^2 x)}{k^2}$ for various values of $N$.
Properties
Riemann function is continuous
Riemann function is almost nowhere differentiable
