# ApartSquareFree

ApartSquareFree[expr]

rewrites a rational expression as a sum of terms whose denominators are powers of square-free polynomials.

ApartSquareFree[expr,var]

treats all variables other than var as constants.

# Details and Options

• ApartSquareFree gives the square-free partial fraction decomposition of a rational expression.
• ApartSquareFree[expr,Trig->True] treats trigonometric functions as rational functions of exponentials, and manipulates them accordingly.
• ApartSquareFree automatically threads over lists in expr, as well as equations, inequalities, and logic functions.

# Examples

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## Basic Examples(1)

Decompose into partial fractions using square-free factorization of the denominator:

Decompose into partial fractions using full factorization of the denominator:

## Scope(7)

### Basic Uses(5)

Decompose a rational function into partial fractions using square-free factorization:

ApartSquareFree can handle symbolic parameters:

Treat as the main variable and as a constant:

Treat as the main variable and as a constant:

Here ApartSquareFree picks as the main variable and treats as a constant:

ApartSquareFree can handle non-polynomial expressions:

ApartSquareFree threads over equations and inequalities:

Compute the square-free partial fraction representation over the integers modulo :

Compute the square-free partial fraction representation following common trigonometric identities:

## Options(2)

### Modulus(1)

Partial fraction decomposition over the rationals:

Partial fraction decomposition over the integers modulo 2:

### Trig(1)

Partial fraction decomposition of a trigonometric expression:

## Properties & Relations(1)

Together acts as an inverse of ApartSquareFree:

Wolfram Research (2007), ApartSquareFree, Wolfram Language function, https://reference.wolfram.com/language/ref/ApartSquareFree.html.

#### Text

Wolfram Research (2007), ApartSquareFree, Wolfram Language function, https://reference.wolfram.com/language/ref/ApartSquareFree.html.

#### CMS

Wolfram Language. 2007. "ApartSquareFree." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ApartSquareFree.html.

#### APA

Wolfram Language. (2007). ApartSquareFree. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ApartSquareFree.html

#### BibTeX

@misc{reference.wolfram_2022_apartsquarefree, author="Wolfram Research", title="{ApartSquareFree}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ApartSquareFree.html}", note=[Accessed: 04-February-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_apartsquarefree, organization={Wolfram Research}, title={ApartSquareFree}, year={2007}, url={https://reference.wolfram.com/language/ref/ApartSquareFree.html}, note=[Accessed: 04-February-2023 ]}