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MathGroup Archive 2007

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Re: Difficulties with Complex-Modulus Series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72730] Re: Difficulties with Complex-Modulus Series
  • From: Jean-Marc 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*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.
> 

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 built-in 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,
Jean-Marc

[1] http://documents.wolfram.com/mathematica/functions/Abs


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