Difference between revisions of "Weierstrass nowhere differentiable function"
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The Weierstrass function is | The Weierstrass function is | ||
| − | $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^ | + | $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^k\pi x),$$ |
where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$. | where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$. | ||
=Properties= | =Properties= | ||
| − | + | [[Weierstrass function is continuous]]<br /> | |
| − | + | [[Weierstrass function is nowhere differentiable]]<br /> | |
| − | + | ||
| − | + | =Videos= | |
| − | + | [https://www.youtube.com/watch?v=pCEFZk9Vihs Weierstrass example (17 October 2013)]<br /> | |
| − | + | ||
| + | =References= | ||
| + | [http://kryakin.org/at/hardy_1916_W.pdf] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 17:54, 25 June 2017
The Weierstrass function is $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^k\pi x),$$ where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.
Properties
Weierstrass function is continuous
Weierstrass function is nowhere differentiable
Videos
Weierstrass example (17 October 2013)
