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Math Help - 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:


    \mathfrak{ Weierstrass \ function:}


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


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


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



    Thank you.
    Last edited by Also sprach Zarathustra; June 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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