       fractional derivative (order t) of (Log[x])^n and Log[Log[x]] etc.?

• To: mathgroup at smc.vnet.net
• Subject: [mg91265] fractional derivative (order t) of (Log[x])^n and Log[Log[x]] etc.?
• From: hanrahan398 at yahoo.co.uk
• Date: Tue, 12 Aug 2008 04:47:10 -0400 (EDT)

```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.