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

• To: mathgroup at smc.vnet.net
• Subject: [mg91279] Re: fractional derivative (order t) of (Log[x])^n and Log[Log[x]]
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 13 Aug 2008 04:39:53 -0400 (EDT)
• References: <g7rim5$ice$1@smc.vnet.net>

Hi,

click on any formula at
<http://functions.wolfram.com/ElementaryFunctions/Log/20/03/>
and you will find the Mathematica input.
For the first formula it is
D[Log[1 + z], {z, \[Alpha]}] == z^(1 - \[Alpha]) 
Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], -z]

Regards
   Jens

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.
> 
> (Ideally I would like a formula where t can be any real number, but
> one for t>=0 would be most helpful!)
> 
> The second thing I am seeking is a formula for the fractional
> derivative w.r.t. x (order t) of Log[Log[x]], Log[Log[Log[x]]], etc.,
> and more generally, of Log[...[Log[Log[x]]...], where there are n
> nested log functions, where n is of course a positive integer.
> 
> (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!!
> 
> Basically I need formulae for the order-t fractional derivatives of
> (Log[x])^n and of Log[Log[x]], Log[Log[Log[x]]] (and generally with n
> nested logs), which I can use for variable x and given values of n and
> t, and can also evaluate at given values of x.
> 
> Many thanks in advance.
> 
> Michael
>