Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
1992
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 1992

[Date Index] [Thread Index] [Author Index]

Search the Archive

RE: Partial Fraction Decomposition with imaginary coeff.

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: RE: Partial Fraction Decomposition with imaginary coeff.
  • From: HAY at leicester.ac.uk
  • Date: Mon, 9 MAR 92 12:03:23 GMT

Jason C. Breckenridge, Department of Physics, University of Western Ontario,

Internet:jbrecken at hydra.uwo.ca
writes
 
> I am experiencing some difficulty getting Mathematica to factor
> something like 
>          (x^2 + 1) into (x + I) (x - I).
> I have studied the book, and cannot find anything subtle, let
> alone obvious, which deals with this.



A  FIRST  TRY AT THE FACTORISING :

FactorComplex[expr_,x_] := Times@@(x - (x/.Solve[expr == 0, x]))

 

TESTS:

FactorComplex[x^2 +1, x]

(-I + x) (I + x)

FactorComplex[(x^2 +1)^2, x]

        2        2
(-I + x)  (I + x)

FactorComplex[x^3 +x^2 -2x +1, x] (*Wo

                1/3                            1/3
 1           7 2              (-47 + Sqrt[837])
(- - ---------------------- - -------------------- + x) 
 3                      1/3             1/3
     3 (-47 + Sqrt[837])             3 2
 
                             1/3                            1/3
   1   I                 -7 2              (-47 + Sqrt[837])
  (- - - Sqrt[3] (---------------------- + --------------------) + 
   3   2                             1/3             1/3
                  3 (-47 + Sqrt[837])             3 2
 
               1/3                            1/3
            7 2              (-47 + Sqrt[837])
    ---------------------- + --------------------
                       1/3             1/3
    3 (-47 + Sqrt[837])             3 2
    --------------------------------------------- + x)

                          2
 
                             1/3                            1/3
   1   I                 -7 2              (-47 + Sqrt[837])
  (- + - Sqrt[3] (---------------------- + --------------------) + 
   3   2                             1/3             1/3
                  3 (-47 + Sqrt[837])             3 2
 
               1/3                            1/3
            7 2              (-47 + Sqrt[837])
    ---------------------- + --------------------
                       1/3             1/3
    3 (-47 + Sqrt[837])             3 2
    --------------------------------------------- + x)
                          2

  
  

Together[Expand[%32]]

           2    3
1 - 2 x + x  + x




PROBLEM:

Consider

Expand[(x-1)(  x^5 +x^2 -2x +1)]

              2    3    5    6
-1 + 3 x - 3 x  + x  - x  + x

The following cannot be done since Solve is stumped by  -1 + 3x - 3x^2 + x^3 -

x^5 + x^6 == 0

FactorComplex[-1 + 3x - 3x^2 + x^3 - x^5 + x^6, x]

ReplaceAll::rmix: 
                                                     5
   Elements of {{x -> 1}, ToRules[Roots[1 + <<2>> + x  == 0, x]]}
     are a mixture of lists and non-lists.

ReplaceAll::rmix: 
                                                     5
   Elements of {{x -> 1}, ToRules[Roots[1 + <<2>> + x  == 0, x]]}
     are a mixture of lists and non-lists.

                                              2    5
-(x (x /. {{x -> 1}, ToRules[Roots[1 - 2 x + x  + x  == 0, x]]}))

  
  

A WAY OUT

One way is to arrange for a numerical solution to the offending part.
For this it is conveient to introduce a variant of Solve

SolveN[arg__] := Solve[arg]/.e_ToRules :> N[e];

Now define

FactorComplexN[expr_,x_] := Times@@(x - (x/.SolveN[expr == 0, x]))

 

TEST:

FactorComplexN[-1 + 3x - 3x^2 + x^3 - x^5 + x^6, x]

(-1 + x) (-0.661809 - 0.256023 I + x) (-0.661809 + 0.256023 I + x)

 
  (-0.050841 - 1.17931 I + x) (-0.050841 + 1.17931 I + x) (1.4253 + x)

(-1 + x) (-0.661809 - 0.256023 I + x) (-0.661809 + 0.256023 I + x)

 
  (-0.050841 - 1.17931 I + x) (-0.050841 + 1.17931 I + x) (1.4253 + x)

Expand[%]//Chop

                 2       3       5    6
-1. + 3. x - 3. x  + 1. x  - 1. x  + x

From
Allan Hayes
Department of Mathematics
The University
Leicester LE1 7RH
U.K.
hay at leicester.ac.uk





  • Prev by Date: Symbolic integration of piecewise defined functions
  • Next by Date: Self destructing definitions
  • Previous by thread: Re: Partial Fraction Decomposition with imaginary coeff.
  • Next by thread: Re: Partial Fraction Decomposition with imaginary coeff.