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 >