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Re: Difficulties with ComplexModulus Series
 To: mathgroup at smc.vnet.net
 Subject: [mg72730] Re: Difficulties with ComplexModulus Series
 From: JeanMarc Gulliet <jeanmarc.gulliet at gmail.com>
 Date: Wed, 17 Jan 2007 06:08:34 0500 (EST)
 Organization: The Open University, Milton Keynes, UK
 References: <eohtrs$q48$1@smc.vnet.net>
carlos at colorado.edu wrote:
> Say I have r = (2*I+x)/(2*Ix), 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.
>
Hi Carlos,
What version of Mathematica and system platform were you using when you
performed your tests?
I ask because I have been unsuccessful in my attempts to get any
derivatives of the function *Abs*.
In[1]:=
$Version
r = (2*I + x)/(2*I  x);
R = Abs[r];
Series[R, {x, 0, 4}]
Series[R, {x, Infinity, 4}]
Out[1]=
5.2 for Microsoft Windows (June 20, 2005)
Out[4]=
2 I + x
Abs[]
2 I  x
Out[5]=
2 I + x
Abs[]
2 I  x
As we can see, both calls of the *Series* function only returned the
callee function.
Now, the builtin Mathematica function *Abs* is meant to work with
numeric argument only: "Abs[z] is left unevaluated if z is not a numeric
quantity [1]." Since r is not a numeric expression, the Abs[r] is left
untouched and especially there is no attempt to simplify or transform r.
In[6]:=
NumericQ[r]
Out[6]=
False
The closest thing to your result I could get is by differentiating R
w.r.t. x
In[7]:=
D[R, x]
Out[7]=
1 2 I + x 2 I + x
( + ) Abs'[]
2 I  x 2 2 I  x
(2 I  x)
Substituting a numeric value for x, we get
In[8]:=
% /. x > 2
Out[8]=
1
() Abs'[I]
2
The above result should have been zero. If we nudge Mathematica to
simplify the result, we get ride of the derivative of a constant
(possibly evaluated to one) but still have some value with a change of sign.
In[9]:=
ComplexExpand[%]
Out[9]=
1

2
Is this a bug or a feature, I don't know: at this point, I just gave up!
Best regards,
JeanMarc
[1] http://documents.wolfram.com/mathematica/functions/Abs
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