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Re: Re: partial fraction
*To*: mathgroup at smc.vnet.net
*Subject*: [mg32917] Re: [mg32881] Re: [mg32865] partial fraction
*From*: Murray Eisenberg <murraye at attbi.com>
*Date*: Tue, 19 Feb 2002 02:29:57 -0500 (EST)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <200202181022.FAA14858@smc.vnet.net>
*Reply-to*: murray at math.umass.edu
*Sender*: owner-wri-mathgroup at wolfram.com
What's wrong with the following:
Integrate[1/(1 - x^2)^2, x]
-x/(2*(-1 + x^2)) - Log[-1 + x]/4 + Log[1 + x]/4
Or, if you want to see the partial fractions, using the built-in Apart:
Apart[1/(1 - x^2)^2]
1/(4*(-1 + x)^2) - 1/(4*(-1 + x)) + 1/(4*(1 + x)^2) + 1/(4*(1 + x))
Integrate[%, x]
-1/(4*(-1 + x)) - 1/(4*(1 + x)) - Log[-1 + x]/4 + Log[1 + x]/4
(Or, as you show, then use Map[Integrate[#, x]&, %] to make sure
Mathematica is really doing the integration term-by-term rather than
first somehow simplifying the partial fraction expression.)
BobHanlon at aol.com wrote:
>
> In a message dated 2/16/02 4:45:52 AM, kyguan at hotmail.com writes:
>
> >Do you have any ideas on how to integrate the following function by partial
> >
> >fraction?
> >
> >1/(1-x^2)^2 (Sorry it was supposed to be a minus instead of plus)
> >
> >I have found the answer by using 'the integrator'. However, I have no single
> >
> >clue on how the answer were calculated.
> >
>
> Although it does not appear to be documented, the standard add-on
> package DiscreteMath`RSolve` defines the function PartialFractions.
>
> Needs["DiscreteMath`RSolve`"];
>
> f[x_] := 1/(1-x^2)^2;
>
> pfe1 = PartialFractions[f[x], x]
>
> 1/(4*(-1 + x)^2) - 1/(4*(-1 + x)) + 1/(4*(1 + x)^2) +
> 1/(4*(1 + x))
>
> Integrating term-by-term
>
> Integrate[#,x]& /@ pfe1
>
> -1/(4*(-1 + x)) - 1/(4*(1 + x)) - Log[-1 + x]/4 +
> Log[1 + x]/4
>
> %//Simplify
>
> ((-2*x)/(-1 + x^2) - Log[-1 + x] + Log[1 + x])/4
>
> You could also do the partial fraction expansion yourself
>
> Factor[f[x]]
>
> 1/((-1 + x)^2*(1 + x)^2)
>
> The partial fraction is then of the form
>
> pfe2 = {a, b, c, d}.(1/{x+1, (x+1)^2 , (x-1), (x-1)^2})
>
> d/(-1 + x)^2 + c/(-1 + x) + b/(1 + x)^2 + a/(1 + x)
>
> eqn = Thread[
> (cl =
>
> CoefficientList[Numerator[Together[pfe2]],x])==
> PadRight[
> CoefficientList[Numerator[f[x]],x],
> Length[cl]]]
>
> {a + b - c + d == 1, -a - 2*b - c + 2*d == 0,
>
> -a + b + c + d == 0, a + c == 0}
>
> soln = Solve[eqn, {a,b,c,d}][[1]]
>
> {a -> 1/4, b -> 1/4, c -> -1/4, d -> 1/4}
>
> (pfe2 /. soln) == pfe1
>
> True
>
> Bob Hanlon
> Chantilly, VA USA
--
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
Amherst, MA 01375
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