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