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Re: Simplification to Partial Fraction

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59730] Re: Simplification to Partial Fraction
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 19 Aug 2005 04:31:52 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <ddurbr$oeh$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <ddurbr$oeh$1 at smc.vnet.net>,
 "Jon Palmer" <Jonathan.palmer at new.ox.ac.uk> wrote:

> I was wondering if someone can help with a Partial Fraction problem.
> 
> I have a calculated expression, u, which is a quotient of two polynomials in
> three variables x, y & z.
> 
> 
> u = P(x,y,z)/Q(x,y,z)
> 
> 
> I know that the quotient, when simplified, is a sum of partial fractions of
> the form
> 
> u = R(x,y,z) + S(x,y,z)/(x^2 +y^2)  + T(x,y,z)/(y^2 +z^2) + U(x,y,z)/(z^2
> +x^2)
> 
> 
> Is there a way to simplify the expression into the parial fraction form?

I would expect that, in general, the answer is not unique -- without 
certain requirements on R, S, T, and U. 

For example, starting with

  start = (x - y + z)/(x^2 + z^2) + x*y*z*(x + y + z) +   
     (y^2 + z^2)/(x^2 + y^2) + (x^2 + y^2 + z)/(y^2 + z^2)

which is of the 'sum of partial fraction' form, we use Together to write 
this as P(x,y,z)/Q(x,y,z).

  rat = start // Together // ExpandDenominator

Now 

  Apart[rat]

gives the same result as Apart[rat,z] but Apart[rat,x] and Apart[rat,y] 
give different, but equivalent, expressions -- and all are valid forms 
in that they correspond to the template

 R(x,y,z) + S(x,y,z)/(x^2+y^2) + T(x,y,z)/(y^2+z^2) + U(x,y,z)/(z^2+x^2)

but for _different_ R, S, T, U. 

> I have tried various combinations of Simplify, Apart, Collect etc. and can't
> find a method that works. Any help would be much appreciated.

You can group the denominators into the required form using Collect:

 Collect[Apart[rat], {y^2 + z^2, x^2 + y^2, x^2 + z^2}, Factor]

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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