Re: Re: question

*To*: mathgroup at smc.vnet.net*Subject*: [mg74953] Re: [mg74925] Re: question*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 12 Apr 2007 04:55:36 -0400 (EDT)*References*: <evd3ku$5dd$1@smc.vnet.net> <200704110601.CAA02936@smc.vnet.net>

On 11 Apr 2007, at 15:01, Michael Weyrauch wrote: > Hello, > > well, I think your command > >> In[57]:= >> FullSimplify[Log[x+1],ComplexityFunction\[Rule] >> (Count[{#1},_Log,Infinity]&)] >> > > fails, because Log[z] as implemented in Mathematica does not know > the transformation rules back into Hypergeometric functions. If I > do it as > follows under Mathematica 5.2 it works without problems... > > Unprotect[Log]; > > Simplify[Log[x + 1], ComplexityFunction -> > (Count[{#1}, _Log, Infinity] & ), > TransformationFunctions -> > {Log[z_] := Hypergeometric2F1[1, 1, 2, -z + 1]* > (z - 1)}] > > (*output*) > > x*Hypergeometric2F1[1, 1, 2, -x] > > Of course, I have to use Unprotect, since I am adding a rule to an > internal command. > > So, if for some reason I would like to "simplify" e.g. Log[x+1]^2 > it works as exspected > > Simplify[Log[x + 1]^2, ComplexityFunction -> > (Count[{#1}, _Log, Infinity] & ), > TransformationFunctions -> > {Log[z_] := Hypergeometric2F1[1, 1, 2, -z + 1]* > (z - 1)}] > > (*output*) > > x^2*Hypergeometric2F1[1, 1, 2, -x]^2 > > Therefore, I am not sure, if Andrzej Kozlowski is right here? > > Regards Michael > I think I am ;-) The reason why your code works is that by Unprotecting Log and then running Simplify with your new transformation function you have actually managed to modify the built-in definition of Log. Observe that before modification: In[1]:= x*Hypergeometric2F1[1, 1, 2, -x] Out[1]= Log[x + 1] But now let's run your code: Unprotect[Log]; In[3]:= Simplify[Log[x + 1]^2, ComplexityFunction -> (Count[{#1}, _Log, Infinity] & ), TransformationFunctions -> {Log[z_] := Hypergeometric2F1[1, 1, 2, -z + 1]* (z - 1)}] Out[3]= x^2*Hypergeometric2F1[1, 1, 2, -x]^2 It loks like you have achieved your purpose, but now try this: Log[z] (z - 1)*Hypergeometric2F1[1, 1, 2, 1 - z] and, of course x*Hypergeometric2F1[1, 1, 2, -x] x*Hypergeometric2F1[1, 1, 2, -x] Doesn't look like a very desirabe situation to me. If one wants to restore the original behaviour of Log one needs to Clear it. But here is a nice puzzle (which I will leave to others to amuze themselves with ;-)) The puzzle is that Log is still Protected even though we never used Protect only Unprotect: Attributes[Log] {Listable,NumericFunction,Protected,ReadProtected} so we unprotect it again: Unprotect[Log]; and then Clear[Log] restores the orignal behaviour: Log[x] Log[x] x*Hypergeometric2F1[1, 1, 2, -x] Log[x + 1] Andrzej Kozlowski

**References**:**Re: question***From:*"Michael Weyrauch" <michael.weyrauch@gmx.de>