       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)
(- - ---------------------- - -------------------- + x)
3                      1/3             1/3
3 (-47 + Sqrt)             3 2

1/3                            1/3
1   I                 -7 2              (-47 + Sqrt)
(- - - Sqrt (---------------------- + --------------------) +
3   2                             1/3             1/3
3 (-47 + Sqrt)             3 2

1/3                            1/3
7 2              (-47 + Sqrt)
---------------------- + --------------------
1/3             1/3
3 (-47 + Sqrt)             3 2
--------------------------------------------- + x)

2

1/3                            1/3
1   I                 -7 2              (-47 + Sqrt)
(- + - Sqrt (---------------------- + --------------------) +
3   2                             1/3             1/3
3 (-47 + Sqrt)             3 2

1/3                            1/3
7 2              (-47 + Sqrt)
---------------------- + --------------------
1/3             1/3
3 (-47 + Sqrt)             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

```

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