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Re: Limit, Series and O

• To: mathgroup at smc.vnet.net
• Subject: [mg14612] Re: Limit, Series and O
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 4 Nov 1998 13:46:44 -0500
• Organization: University of Western Australia
• References: <71bkon$pua@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

RENZONI_FERRUCCIO wrote:

> I work in the complex plane, but all my variable are real. It has soon
> become clear that the answer I get running the commands Limit, Series
> and O[x]^k are not correct because I can't define reality and
> (sometimes) positivity of my variables. So for example if I have "a"
> positive and I have as a result of a computation
> 
>  x / ( a - Sqrt[a^2] + x) + O[x]^2
> 
> and I ask to Simplify, I don't get the result I want.

The result you "want" is, of course 

 In[1]:= (x / ( a - Sqrt[a^2] + x)//PowerExpand) + O[x]^2
 Out[1]=
	        2
	1 + O[x]

and, as you've observed, you don't get what you want if you change the
evaluation order:

 In[2]:= x / ( a - Sqrt[a^2] + x) + O[x]^2//PowerExpand
 Out[2]=
	                        2
	ComplexInfinity x + O[x]

So, the moral of this story is that you must algebraically simplify your
expression before you compute the series.  Basically, the series
expansion and simplification do not commute.

> So I am trying to "redefine" the various commands "Sqrt,Log,Power..." 

I cannot see how this will resolve your problem.  I understand that you
want these simplifications to take place automatically, but in general
(for many variables), I don't think this is possible.

Cheers,
	Paul


____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at physics.uwa.edu.au  AUSTRALIA                       
http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________