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
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