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MathGroup Archive 2007

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Re: Difficulties with Complex-Modulus Series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72753] Re: [mg72712] Difficulties with Complex-Modulus Series
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 17 Jan 2007 07:41:00 -0500 (EST)
  • References: <200701160803.DAA29016@smc.vnet.net>

The absolute value function is not differentiable so Mathematica  
naturally does not know what its derivative might be. The evaluator  
does not automatically perform any but the simplest simplifications.  
But all you need to do to avoid this problem is:


r = (2*I+x)/(2*I-x);R=Abs[r];

In[4]:=
Series[ComplexExpand[R],{x,0,4}]

Out[4]=
1

In[5]:=
Series[ComplexExpand[R],{x,Infinity,4}]

Out[5]=
1

Not too much to ask, it seems to me.

Moral: computer programs and even Mathematica, are not human, yet.

Andrzej Kozlowski




On 16 Jan 2007, at 09:03, carlos at colorado.edu wrote:

> Say I have  r = (2*I+x)/(2*I-x),  in which x is real and nonnegative.
>
> Series[r,{x,0,4}]  and  Series[r,{x,Infinity,4}]   work as expected.
>
> Introduce now  R=Abs[r] and try the same:
>
> Series[R,{x,0,4}]  and  Series[R,{x,Infinity,4}]
>
> Results are now "contaminated" with Abs'[-1], Abs''[-1], etc,
> I dont understand the presence of those derivatives.
> Anybody can explain the reason?  (I teach students that the
> derivative of a constant is zero, but perhaps that has changed
> with the new year)  BTW it would be nice to say
>
> Series[R,{x,0,4}, x>=0]  or Series[R,{x,0,4}, R>=0]  etc
>
> if that would get rid of the garbage,  but Mathematica 5
> does not allow Assumptions in Series. Note BTW that R=1 for
> any x, so the R series are in fact trivial to any order.
>


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