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MathGroup Archive 2007

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Re: Apart question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73046] Re: Apart question
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 30 Jan 2007 23:56:39 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <epn9uv$cn3$1@smc.vnet.net>

In article <epn9uv$cn3$1 at smc.vnet.net>,
 "dimitris" <dimmechan at yahoo.com> wrote:

> Dear All,
> 
> In[317]:=
> f[x_]:=(x^2+2*x+4)/(x^4-7*x^2+2*x+17)
> 
> In[323]:=
> Apart[f[x]]
> 
> Out[323]=
> (4 + 2*x + x^2)/(17 + 2*x - 7*x^2 + x^4)
> 
> In[320]:=
> Times@@Apply[#1[[1]] - #1[[2]] & , Solve[Denominator[f[x]] == 0, x], 
> 1]
> Apart[(4 + 2*x + x^2)/%]
> Map[FullSimplify, %, 1]
> 
> Out[320]=
> (I/2 - (1/2)*Sqrt[15 - 4*I] + x)*(-(I/2) - (1/2)*Sqrt[15 + 4*I] + 
> x)*((1/2)*(I + Sqrt[15 - 4*I]) + x)*
>   ((1/2)*(-I + Sqrt[15 + 4*I]) + x)
> 
> Out[321]=
> (4*I*((4 + 15*I) + (1 + 2*I)*Sqrt[15 - 4*I]))/(Sqrt[15 - 4*I]*(-2*I + 
> Sqrt[15 - 4*I] - Sqrt[15 + 4*I])*
>     (-2*I + Sqrt[15 - 4*I] + Sqrt[15 + 4*I])*(-I + Sqrt[15 - 4*I] - 
> 2*x)) - (4*I*((-4 - 15*I) + (1 + 2*I)*Sqrt[15 - 4*I]))/
>    (Sqrt[15 - 4*I]*(2*I + Sqrt[15 - 4*I] - Sqrt[15 + 4*I])*(2*I + 
> Sqrt[15 - 4*I] + Sqrt[15 + 4*I])*(I + Sqrt[15 - 4*I] + 2*x)) -
>   (4*((15 + 4*I) + (2 + I)*Sqrt[15 + 4*I]))/(Sqrt[15 + 4*I]*(-2*I + 
> Sqrt[15 - 4*I] - Sqrt[15 + 4*I])*
>     (2*I + Sqrt[15 - 4*I] + Sqrt[15 + 4*I])*(-I - Sqrt[15 + 4*I] + 
> 2*x)) + (4*((-15 - 4*I) + (2 + I)*Sqrt[15 + 4*I]))/
>    (Sqrt[15 + 4*I]*(-2*I - Sqrt[15 - 4*I] + Sqrt[15 + 4*I])*(-2*I + 
> Sqrt[15 - 4*I] + Sqrt[15 + 4*I])*(-I + Sqrt[15 + 4*I] + 2*x))
> 
> Out[322]=
> 1/(1 + Sqrt[-15 + 4*I] - 2*I*x) + 1/(1 - Sqrt[-15 - 4*I] + 2*I*x) + 1/
> (1 + Sqrt[-15 - 4*I] + 2*I*x) -
>   1/(-1 + Sqrt[-15 + 4*I] + 2*I*x)
> 
> In[323]:=
> Options[Apart]
> 
> Out[323]=
> {Modulus -> 0, Trig -> False}
> 
> Why Apart cannot provide straightly the output Out[322]?

Why does Factor not factor the denominator into linear factors? With the 
appropriate extension (Extension -> Automatic does not work), one can 
factor the denominator

  Factor[(x^2 + 2*x + 4)/(x^4 - 7*x^2 + 2*x + 17), 
    Extension -> {Sqrt[-15 + 4*I], Sqrt[-15 - 4*I]}]

and then obtain the desired result using Apart and FullSimplify.

  FullSimplify /@ Apart[%]

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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