PartialFractionElements
PartialFractionElements[expr,x]
gives the data elements related to a partial fraction decomposition of the rational function expr in x.
PartialFractionElements[expr,x,prop]
gives the data elements for property prop.
Details
- Partial fraction decompositions, or partial fraction expansions, have applications in the integration of rational functions and other calculus operations.
- Let
be a rational function of 
with numerator 
and denominator 
. Then the complete partial fraction decomposition of 
has the form: -
![r(x)=sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript]+sum_(i=1)^msum_(j=1)^(k_i)(a(i,j))/((x-beta_i)^j)](Files/PartialFractionsElements.en/6.png "r(x)=sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript]+sum_(i=1)^msum_(j=1)^(k_i)(a(i,j))/((x-beta_i)^j)")
- where
![sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript]](Files/PartialFractionsElements.en/8.png "sum_(i=1)^pgamma_i TemplateBox[{x, i}, Superscript]")
is the polynomial part of the decomposition, 
are the poles of 
(i.e. the roots of 
) with multiplicities 
, and 
is the fractional part of the decomposition. - PartialFractionElements returns an Association with data related to the above partial fraction decomposition of
, as determined by PartialFractions. The returned data is of the following two kinds: - Data related to the algebraic expression for the partial fraction decomposition:
-
"Decomposition" 
"FractionalPart" 
"PolynomialPart" 
- Data related to the coefficients appearing in the partial fraction decomposition:
-
"FractionalCoefficients" 
"PoleMultiplicities" 
"Poles" 
"PolynomialCoefficients" 
Examples
open all close allScope (1)
Compute all elements for a partial fraction decomposition:
Compute just the decomposition:
Check that it is equivalent to the rational function input:
Get the polynomial part of the decomposition:
Obtain the lists of {pole,mult,coeffs} with the coefficients ordered from degree 1 to the full multiplicity for each pole:
Similarly, obtain the coefficient list for the polynomial part:
Text
Wolfram Research (2026), PartialFractionElements, Wolfram Language function, https://reference.wolfram.com/language/ref/PartialFractionsElements.html.
CMS
Wolfram Language. 2026. "PartialFractionElements." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PartialFractionsElements.html.
APA
Wolfram Language. (2026). PartialFractionElements. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PartialFractionsElements.html