Difference between revisions of "Faber F2"
From specialfunctionswiki
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The Faber function $F_2$ is defined by | The Faber function $F_2$ is defined by | ||
$$F_2(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!}x-m \right|.$$ | $$F_2(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!}x-m \right|.$$ | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Faberf2plot.png|Graph of $F_2$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The Faber function $F_2$ is [[continuous]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The Faber function $F_2$ is [[nowhere differentiable]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=See Also= | =See Also= | ||
Revision as of 19:38, 22 January 2016
The Faber function $F_2$ is defined by $$F_2(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!}x-m \right|.$$
Graph of $F_2$.
Properties
Theorem: The Faber function $F_2$ is continuous.
Proof: █
Theorem: The Faber function $F_2$ is nowhere differentiable.
Proof: █
