Re: Uniform design

*To*: mathgroup at smc.vnet.net*Subject*: [mg48365] Re: Uniform design*From*: ab_def at prontomail.com (Maxim)*Date*: Tue, 25 May 2004 07:18:08 -0400 (EDT)*References*: <c8pu2d$kq9$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Continuing on the topic of uniform design, let's see how well various functions handle SeriesData. For instance, series expansions can be integrated just like other expressions: In[1]:= Integrate[Log[a] + x + O[a], {x, 0, 1}] Out[1]= a + O[a]^2 The result is incorrect, probably because Integrate overlooks the possibility that SeriesData can contain logarithmic terms. Next, we can substitute values into series: In[2]:= z + O[n] /. z -> 1/n + 1 z + O[n]^2 /. z -> 1/n + 1 Out[2]= 1 + O[n] Out[3]= 1/n + 1 + O[n]^3 Only in the first case we lose the 1/n term, in the second case lose O[n]^2 term. Let's look at Limit: In[4]:= Limit[O[n]/n^2, n -> 0] Out[4]= ComplexInfinity Strictly speaking, this is incorrect too, because n^2 is also O[n]. Another issue is that Mathematica outputs O[n]/n^2 as 1/O[n], which I suppose can be acceptable as a conventional notation, but it's questionable in the strict mathematical sense: O(n)/n^2 is O(1/n) and not 1/O(n). Limit of 1/O(n) (not Mathematica's 1/O[n]!) would indeed be equal to ComplexInfinity. Another example: In[5]:= Limit[z/n + O[n], n -> 0] Out[5]= ComplexInfinity At least this one is good? Not exactly, because standard Mathematica behaviour in such cases is to return z*Infinity. In[6]:= HarmonicNumber[n, 2] + O[n, Infinity] -HarmonicNumber[n, 2] + O[n, Infinity] Out[6]= Pi^2/6 + O[1/n]^1 Out[7]= -HarmonicNumber[n, 2] + O[1/n]^1 The last example demonstrates a different issue: transformation rules catch only some of all the possible cases; here changing the sign prevents the application of the rule, sometimes f[n]+O[n] and Series[f[n],{n,0,0}] work slightly differently. Lastly, there are some other problems with notation: O[x]^0 is allowed in the output, but not as the input; O[1/x] will be output exactly as O[x,Infinity] but means something different. Maxim Rytin m.r at inbox.ru (don't use the prontomail address anymore please)

**Follow-Ups**:**Re: Re: Uniform design***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>