Re: Partial Fractions
- To: mathgroup at smc.vnet.net
- Subject: [mg27719] Re: [mg27709] Partial Fractions
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Tue, 13 Mar 2001 03:52:38 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
It would be better if you provided an example. Mathematica's Apart is definitely incomplete. If you want to expand expressions into partial fractions over the complex numbers you can download the package: <http://www.mathsource.com/MathSource/Enhancements/Algebraic/0202-071/MyApar t.m> For example with the built in mathematica Apart you get: In[1]:= Apart[1/(1+x^4)] Out[1]= 1 ------ 4 1 + x with the MyApart package: In[2]:= MyApart[1/(1+x^4),x] Out[2]= 3/4 (-1) -(--------------------------------------------------------) + 1/4 3/4 1/4 3/4 1/4 2 ((-1) - (-1) ) ((-1) + (-1) ) (-(-1) + x) 3/4 (-1) --------------------------------------------------------- + 1/4 3/4 1/4 3/4 1/4 2 (-(-1) - (-1) ) (-(-1) + (-1) ) ((-1) + x) 1/4 (-1) ---------------------------------------------------------- - 1/4 3/4 1/4 3/4 3/4 2 (-(-1) - (-1) ) (-(-1) + (-1) ) (-(-1) + x) 1/4 (-1) --------------------------------------------------------- 1/4 3/4 1/4 3/4 3/4 2 (-(-1) - (-1) ) (-(-1) + (-1) ) ((-1) + x) However, you may prefer the expansion over the extension of the rationals by Sqrt[2] (unfortunately, unlike several other algebraic functions Apart does not accept the Extension option). One quick way to get this is as follows: In[3]:= D[Integrate[1/(1 + x^4), x], x] Out[3]= Sqrt[2] - 2 x Sqrt[2] + 2 x -(-------------------------------) + ------------------------------ + 2 2 4 Sqrt[2] (-1 + Sqrt[2] x - x ) 4 Sqrt[2] (1 + Sqrt[2] x + x ) 1 1 --------------------------- + -------------------------- 1 2 1 2 2 (1 + - (-Sqrt[2] + 2 x) ) 2 (1 + - (Sqrt[2] + 2 x) ) 2 2 checking: In[4]:= Simplify[%] Out[4]= 1 ------ 4 1 + x on 3/12/01 8:09 AM, Bruce Detterich at bruce.detterich at ieee.org wrote: > "The Book" and other sources shows Partial Fraction expansion using > sequences of commands, usually ending in Apart[]. However, I've > encountered a few cases that seem as though they should go fairly > gracefully to Partial Fractions, but no amount of fiddling seems to > make it happen. This includes taking individual components and > working at them. So, has the Partial Fraction expansion procedure > been captured anywhere? > >