       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
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:=
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=

\[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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