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Re: Abs and variables

• To: mathgroup at smc.vnet.net
• Subject: [mg4297] Re: Abs and variables
• From: Mark Evans <evans at gte.net>
• Date: Sat, 29 Jun 1996 03:56:36 -0400
• Organization: GTE Intelligent Network Services, GTE INS
• Sender: owner-wri-mathgroup at wolfram.com

Perrin wrote:
> 
>         How can I have the Abs function treat real variables properly?
> I seem to recall a way of declaring a variable real, but I don't
> remember how.  If there is such a way, would this work with Abs; i.e.
> fix the variable a so that "Abs[a]" will give a result of "a"?
> 
> Thanks.
> 
> /
> :@-) Scott
> \

I have always bemoaned Mathematica's deficiency with respect to symbolic real 
variables.  WRI has vaguely promised some improvements in this area.  I'm 
sure they will do a better job next time around.

Mathematica is just not very good at simplifying symbolic expressions 
involving symbolic reals.  The kernel always assumes that symbols could be 
complex even though in many cases they cannot.

The particular simplification you cite, however, is not quite valid in any 
case, because "a" can be positive or negative even if it is real.  If a < 0, 
then Abs[a] != a.  In this case I would not want the simplification you cite 
to take effect.

There is a package called "ReIm.m" which lets you declare symbols to be pure 
reals, but I have not found it very helpful.  One never seems to get the 
straightforward simplifications that one would expect in reasonable amounts 
of time.

There is also the ComplexExpand[] function.  This is the most helpful 
function I have found in this regard.  Read the manual about its usage.  When 
you wrap an expression with ComplexExpand[], the symbols in the expression 
are assumed real unless you specifically state otherwise in the argument list 
to the function.

Mark Evans
evans at gte.net

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