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MathGroup Archive 2004

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Re: RE : Bug in O[x]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48283] Re: [mg48264] RE : [mg48233] Bug in O[x]
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Fri, 21 May 2004 03:54:31 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200405200803.EAA16147@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

This thread reminds me of the discrepancy between how Mathematica treats 
O[x^n] and the most common definition of this in mathematics.  In 
mathematics, O[x^n} ordinarily denotes some function whose magnitude is 
bounded by a constant times Abs[x^n]  (for suitably large magnitude 
values of x).  In particular, O[x} is a function whose magnitude is 
bounded by c Abs[x].

Physicists and other such non-mathematician types often use the 
definition you stated, that O[x^] = a x^n for suitable constant a. 
Mathematicians even sometimes say that when they are trying not to be so 
technical.

Note also that Mathematica doesn't seem to know much, or anything, about 
O[x^n}, only O[x]^n.  Thus:

   O[x] + O[x^2}
O[x^2]^2 + O[x]^1

   O[x] + O[x]^2
O[x]

Florian Jaccard wrote:

> ... You have to consider O[x] as somethig of the form a*x , O[x]^2 as something
> of the form a*x^n , etc.
> Rule :	O[x]+O[x^2]=O[x]  etc.
> 
> (1+O[x])^2=1+2*O[x]+O[x]^2=1+2*O[x]=1+O[x]
> 
> (1/x+O[x])^2=1/x^2+2*1/x*O[x]+O[x]^2=1/x^2+2*1/x*O[x]=x^(-2)+O[x]^0
> (because 1/x * O[x]  is something behaving like a constant...)
> 
> (1+O[x])^3=1+3*O[x]+3*O[x]^2+O[x]^3=1+O[x]
> 
> Nothing is wrong... you just d'd'nt understand the meaning of O[x]...
> 
> O[x]^n represents a term of order x^n. O[x]^n is generated to represent \
> omitted higher-order terms in power series.

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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