Re: Arctangent approximation
- To: mathgroup at smc.vnet.net
- Subject: [mg107578] Re: [mg107560] Arctangent approximation
- From: Murray Eisenberg <murray at math.umass.edu>
- Date: Thu, 18 Feb 2010 05:15:37 -0500 (EST)
- Organization: Mathematics & Statistics, Univ. of Mass./Amherst
- References: <201002171202.HAA21864@smc.vnet.net>
- Reply-to: murray at math.umass.edu
I'll buy the first four terms of the approximation but not the fifth.
What you seem to have here are the first 10 terms -- the first 5 nonzero
terms -- of the Taylor expansion of ArcTan[x] around 0.
Mathematica gives the result as:
Series[ArcTan[x], {x, 0, 9}] // Normal
The result is a standard one that is taught in any standard calculus
course. You obtain it by beginning with the geometric series expansion
(1-x)^(-1) = Sum[x^n, n,0,Infinity]
(valid for Abs[x]<1), substitute there x = -(x^2), integrate the result
term-by-term (which is legitimate for Abs[x] < 1), and then take the
first 9 terms of that resulting infinite series.
On 2/17/2010 7:02 AM, sidey wrote:
> A professor (Herbert Medina) came up with this remarkable
> approximation.
>
> Since I don't have Mathematica, could someone run it for
> numberofdigits=9 and 12 please? (and send me the result?)
>
> here is a short result
> h2(x) = x - x^3/3 + x^5/5 - x^7/7 + 5x^9/48..
>
> Thank you.
>
> PS reference is: http://myweb.lmu.edu/hmedina/Papers/Arctan.pdf
>
> Clear[h, m, a, numberofdigits]
> numberofdigits=12;
> m=Floor[-(1/5) Log[4,5*0.1^(numberofdigits+1)]]+1;
> Do[a[2j]=(-1)^(j+1) Sum[Binomial[4m,2k] (-1)^k, {k,j+1,2m}];
> a[2j-1]=(-1)^(j+1) Sum[Binomial[4m,2k+1] (-1)^k, {k,j,2m-1}],
> {j,0,2m-1}];
> h[m,x_]:=Sum[(-1)^(j+1) / (2j-1) x^(2j-1), {j,1,2m}] +
> Sum[a[j]/((-1)^(m+1) 4^m (4m+j+1)) x^(4m+j+1), {j,0, 4m-2}];
> Print["h[",m,",x] given below computes ArcTan[x] with ",
> numberofdigits," digits of accuracy for x in [0,1]."]
> Print["h[m,x] = ", h[m,x]]
>
--
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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- References:
- Arctangent approximation
- From: sidey <sidey.p.timmins@census.gov>
- Arctangent approximation