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Re: Apart question

• To: mathgroup at smc.vnet.net
• Subject: [mg73051] Re: [mg73030] Apart question
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 31 Jan 2007 00:16:44 -0500 (EST)
• References: <200701301140.GAA14182@smc.vnet.net>

On 30 Jan 2007, at 12:40, dimitris 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]?
>


I am not sure if this answer will satisfy you but one way to put it  
is: for exactly the same reason why Factor requires you to specify  
the extension manually. In fact, if you use Apart with Factor and  
specify the correct extension you will get output equivalent to your  
Out[322]. If, in addition, you FullSimplify the individual terms you  
will get exactly Out[322].

In[50]:=
FullSimplify /@ Apart[Factor[(x^2 + 2*x + 4)/(x^4 - 7*x^2 + 2*x + 17),
     Extension -> {Sqrt[-15 - 4*I], Sqrt[-15 + 4*I]}]]

Out[50]=
1/(2*I*x - Sqrt[-15 - 4*I] + 1) + 1/(2*I*x + Sqrt[-15 - 4*I] + 1) -
   1/(2*I*x + Sqrt[-15 + 4*I] - 1) + 1/(-2*I*x + Sqrt[-15 + 4*I] + 1)

Andrzej Kozlowski

• References:
  • Apart question
    • From: "dimitris" <dimmechan@yahoo.com>