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Re: fractional derivative (order t) of (Log[x])^n and Log[Log[x]]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91288] Re: fractional derivative (order t) of (Log[x])^n and Log[Log[x]]
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 13 Aug 2008 04:41:45 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <g7rim5$ice$1@smc.vnet.net>

hanrahan398 at yahoo.co.uk wrote:

> I'd be grateful if someone could tell me a nicely computable formula
> for the fractional derivative w.r.t. x (order t) of (Log[x])^n, where
> n is a positive integer.

<snip>

> (I have visited:
> <http://functions.wolfram.com/ElementaryFunctions/Log/20/03/>, and
> there is a formula there for the t-th (fractional) derivative of
> Log[x]^n, but I do not understand how to input it!!

<snip>

You may have not notice but if you look at the upper-left corner below 
the graphic, more or less at the same level at the section title 
"Elementary Functions  Log[z]  Differentiation", you can see a paragraph 
titled "DOWNLOAD FORMULAS FOR THIS FUNCTION": just below it you can 
choose to download the file as a Mathematica notebook or as PDF.

http://functions.wolfram.com/ElementaryFunctions/Log/20/03/

Then you can explore and manipulate the formulae directly from within 
Mathematica. For instance,


In[1]:=
FullSimplify[
    D[Log[z], {z, \[Alpha]}] ==
      Piecewise[
        {{((-1)^(\[Alpha] - 1)*
                 (\[Alpha] - 1)!)/z^\[Alpha],
            Element[\[Alpha],
                Integers] &&
              \[Alpha] > 0}},
        (Log[z] - EulerGamma -
               PolyGamma[1 - \[Alpha]])/
            Gamma[1 - \[Alpha]]/z^\[Alpha]]]

Out[1]=

                  \[Alpha]
             -(-1)  Gamma[\[Alpha]]
Piecewise[{{---------------,
                    \[Alpha]
                   z

      \[Alpha] \[Element] Integers && \[Alpha] > 0}},

    -HarmonicNumber[-\[Alpha]] +

      Log[z]
    ----------------------------
           \[Alpha]
          z  Gamma[1 - \[Alpha]]

             (\[Alpha])
     ] == Log   [z]


HTH,
-- Jean-Marc


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