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

**References**:**RE : Bug in O[x]***From:*"Florian Jaccard" <florian.jaccard@eiaj.ch>