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Thread: Differentiability of the Weierstrauss Function

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Differentiability of the Weierstrass Function

    I want to know something about:


    $\displaystyle \mathfrak{ Weierstrass \ function:}$


    $\displaystyle \sum_{n=0}^{\infty}a^n\cos(b^n\pi x)$ where $\displaystyle 0<a<1$ and $\displaystyle b$ is odd integer so $\displaystyle ab>1+\frac{3}{2}\pi$.


    1. How to prove that Weierstrass function is nowhere differentiable?


    2. Why $\displaystyle b$ must be odd? And why $\displaystyle ab>1+\frac{3}{2}\pi$?



    Thank you.
    Last edited by Also sprach Zarathustra; Jun 15th 2011 at 03:57 PM.
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  2. #2
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    Re: Differentiability of the Weierstrauss Function

    For (1), see Weierstrass's paper. For (2), these conditions are artificial and unnecessary. See Hardy's paper on a generalization.
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