Partial fraction expansion

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11.6.9 Partial fraction expansion

The partfrac and cpartfrac commands find the partial fraction expansion of a rational function.

partfrac takes rat, a rational function.
partfrac(rat) returns the partial fraction expansion of rat.
The partfrac command is equivalent to the convert command (see Section 10.1.10) with parfrac (or partfrac or fullparfrac) as option.
cpartfrac(rat) behaves just like partfrac, except that it always finds the partial fraction expansion over ℂ.

Example

Find the partial fraction expansion of x⁵−2x³+1/x⁴−2x³+2x²−2x+1 over the real numbers.

Input in real mode:

partfrac((x^5-2*x^3+1)/(x^4-2*x^3+2*x^2-2*x+1))

x+2−

1

2

⎛
⎝

x−1

⎞
⎠

+

x−3

2

⎛
⎝

x²+1

⎞
⎠

To find the partial fraction decomposition over the complex numbers, you can either put Xcas in complex mode (see Section 2.5.5) or use cpartfrac.

Input in complex mode:

partfrac((x^5-2*x^3+1)/(x^4-2*x^3+2*x^2-2*x+1))

or, in real or complex mode:

cpartfrac((x^5-2*x^3+1)/(x^4-2*x^3+2*x^2-2*x+1))

x+2−

1

2

⎛
⎝

x−1

⎞
⎠

+

−1−2 i

⎛
⎝

2−2 i

⎞
⎠

⎛
⎝

x+i

⎞
⎠

+

2+i

⎛
⎝

2−2 i

⎞
⎠

⎛
⎝

x−i

⎞
⎠

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