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Half Derivative Of (ax+c)^(3/2)

  1. Dec 11, 2011 #1
    d^1/2(ax+c)^(3/2)/dx^(1/2)=? please help me :)
     
  2. jcsd
  3. Dec 11, 2011 #2

    Simon Bridge

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    It is tempting to start by taking everything under the common square root, isn't it?
    However, you have probably figured out that this does not work for the derivative operator.

    Just like when [itex]y=x^{1/2}[/itex], you are actually looking for [itex]y: y^2=x[/itex], when you see [itex]D^{1/2}f[/itex] you are actually looking to use something like [itex]Hf(x) :H^2f(x)=Df(x)[/itex].

    Look for http://en.wikipedia.org/wiki/Fractional_calculus]fractional[/PLAIN] [Broken] calculus[/itex].
     
    Last edited by a moderator: May 5, 2017
  4. Dec 12, 2011 #3
    In using the Riemann-Liouville operator for the definition of the fractionnal derivative, and in order to avoid somme difficulties of integral convergence, it will be simpler to :
    First, compute the fractional integral (order 1/2) of the function.
    Then, compute the classical derivative.
     

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  5. Dec 12, 2011 #4
    Even simpler, by performing first the classical derivation, then the fractionnal integration (joint page)
    (be carefull with this manner, which is generally not recommended. But in the present case, it works)
     

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    Last edited: Dec 12, 2011
  6. Dec 13, 2011 #5
    You're welcome !

    I used fractionnal derivatives in complex impedance analysis.
    The particular case of half derivative is common in modeling the electrical properties of many homogeneous compounds (a particular behaviour and impedance, long ago loosely refreed as Warburg's impedance).
    Especially, the fractionnal derivatives of the sinusoidal functions are interesting.
    For example, in the paper "The Phasance Concept", pp.5-6
    http://www.scribd.com/JJacquelin/documents
     
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