Re: Re: O in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg95997] Re: [mg95950] Re: O in Mathematica
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Sat, 31 Jan 2009 06:45:10 -0500 (EST)
- References: <gls1vl$hl1$1@smc.vnet.net> <200901301046.FAA06594@smc.vnet.net>
- Reply-to: drmajorbob at longhorns.com
> 2 2 > Pi 1 1 8 + 3 Pi 8 + Pi 1 5 > Out[9]= Pi n + -- - ---- + ------- - --------- + -------- + O[-] > 2 Pi n 2 3 3 3 4 n > 2 Pi n 12 Pi n 8 Pi n I gather (just guessing) that's an approximation to n Pi + Arctan[n] for large n? Bobby On Fri, 30 Jan 2009 04:46:44 -0600, Scott Hemphill <hemphill at hemphills.net> wrote: > Francois Fayard <fayard.prof at gmail.com> writes: > >> Hello, >> >> At first, thanks for your help, but I've found what I was asking for. >> To input a O(1) in mathematica, you juste have to write >> >> n O[n,Infinity] >> >> which gives you O[1/n]^0 which is not simplified to 0. >> >> Now, I've got another question around O. Let's first explain what I >> call a O, or big O (in France). A O(f(x)) around zero is a function >> that can be written B(x)f(x) where B(x) is bounded around 0. I just >> want to make sure everyone speaks about the same thing. >> With that definition x = O(x) (around 0), but x Log[x] is not a O(x) >> (around 0) as x Log[x]/x=Log[x] is not bounded around 0. But when I >> write in Mathematica >> >> Log[x] O[x,0]^1 >> >> It is simplified to O[x,0]^1 which is obviously wrong. I've seen that >> if you multiply O[x,0]^1 by a fonction g(x) that is negligeable >> compared to x^epislon around 0 for a epsilon>0, the result is >> simplified to O[x,0]^1 which is wrong form a mathematical point of view. >> >> Do I have to understand that O[x,0]^n (in Mathematica) should be >> considered as a O[x,0]^(n-epsilon) (in mathematics) for whatever >> epsilon>0 you want ? If we consider this definition, are the results >> from Mathematica "certified" ? >> Another question should be : Why does Mathematica behave like that ? > > Hi Francois, > > Did you see my response to one of your other posts? In it I develop > the following expression, which gives the expansion you want to the > number of terms you desire: > > In[8]:= f[k_] := FixedPoint[Simplify[Series[n*Pi + ArcTan[Normal[#1]], > {n, Infinity, k}]] & , Infinity] > > In[9]:= f[4] > > 2 2 > Pi 1 1 8 + 3 Pi 8 + Pi 1 5 > Out[9]= Pi n + -- - ---- + ------- - --------- + -------- + O[-] > 2 Pi n 2 3 3 3 4 n > 2 Pi n 12 Pi n 8 Pi n > > Scott -- DrMajorBob at longhorns.com
- References:
- Re: O in Mathematica
- From: Scott Hemphill <hemphill@hemphills.net>
- Re: O in Mathematica