Re: Re: Abs[x] function

*To*: mathgroup at smc.vnet.net*Subject*: [mg87880] Re: [mg87845] Re: Abs[x] function*From*: "Szabolcs HorvÃt" <szhorvat at gmail.com>*Date*: Sat, 19 Apr 2008 03:34:51 -0400 (EDT)*References*: <fu9fn7$c8d$1@smc.vnet.net> <200804181110.HAA18822@smc.vnet.net>

On Fri, Apr 18, 2008 at 2:14 PM, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > > On 18 Apr 2008, at 20:10, Szabolcs wrote: > > > On 18 Apr, 08:39, Vladislav <kazimi... at yahoo.co.uk> wrote: > > > > > Who can explain the behavior. THe derivative Abs[x] at x=.5 is well > > > defined and is equal to 1. > > > > > > In[1]:= D[Abs[x], x] > > > > > > Out[1]= > > > \!\(\*SuperscriptBox["Abs", "\[Prime]", > > > MultilineFunction->None]\)[x] > > > > > > In[2]:= % /. x -> .5 > > > > > > Out[2]= > > > \!\(\*SuperscriptBox["Abs", "\[Prime]", > > > MultilineFunction->None]\)[0.5] > > > > > > > Hi, > > > > Please use "copy as plain text" when pasting Mathematica expressions, > > so it will be easier to read them. > > > > Just use FunctionExpand on the result to get a concrete value. > > > > > > > I never noticed that Function expand deos that and I am not sure that I am > very happy that it does. In fact: > > FunctionExpand[Derivative[1][Abs][x], > Element[x, Reals]] > x/Abs[x] > > but that assumes that the derivative is taken along the real line, which is > not what Mathematica normally assumes. In my opinion this makes it less > clear exactly what FunctionExpand is really meant to do and what to expect > of it. I don't think it ought to make additional assumptions beyond what it > is told. > > I think it would be better to let Derivative have a direction option, just > as Limit does. Somewhat related to the topic: It turns out that Limit understands Abs'[x] (this comes as a surprise to me): In[1]:= Limit[Abs'[x], x -> 0, Direction -> 1] Out[1]= -1 In[2]:= Limit[Abs'[x], x -> 0, Direction -> -1] Out[2]= 1 But it does not work correctly with complex directions: In[3]:= Limit[Abs'[1 + x], x -> 0, Direction -> I] Out[3]= 1 The result should have been 0. Szabolcs

**References**:**Re: Abs[x] function***From:*Szabolcs <szhorvat@gmail.com>