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Apart

Apart[expr]

rewrites a rational expression as a sum of terms with minimal denominators.

Apart[expr,var]

treats all variables other than var as constants.

Details and Options

  • Apart gives the partial fraction decomposition of a rational expression.
  • Apart[expr,var] writes expr as a polynomial in var together with a sum of ratios of polynomials, where the degree in var of each numerator polynomial is less than that of the corresponding denominator polynomial.
  • Apart[expr,Trig->True] treats trigonometric functions as rational functions of exponentials, and manipulates them accordingly.
  • Apart automatically threads over lists in expr, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (2)Summary of the most common use cases

Decompose into partial fractions:

Compute the partial fraction representation of a rational function:

Scope  (8)Survey of the scope of standard use cases

Basic Uses  (6)

Decompose a rational function into partial fractions:

Apart can handle symbolic parameters:

Treat as the main variable and as a constant:

Treat as the main variable and as a constant:

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

Apart can handle non-polynomial expressions:

Apart threads over equations and inequalities:

Compute the partial fraction representation when radicals are present:

Advanced Uses  (2)

Compute the partial fraction representation over the integers modulo :

Compute the partial fraction representation following common trigonometric identities:

Options  (3)Common values & functionality for each option

Modulus  (2)

Partial fraction decomposition over the rationals:

Partial fraction decomposition over the integers modulo 2:

Trig  (1)

Partial fraction decompositions of trigonometric expressions:

Applications  (1)Sample problems that can be solved with this function

Integrals of rational expressions of polynomials are often computed by decomposing into partial fractions:

To compute the integral, first apply Apart to the rational expression:

Apply Integrate to each summand individually and sum the results:

Verify the answer:

Properties & Relations  (1)Properties of the function, and connections to other functions

Together acts as an inverse of Apart:

Wolfram Research (1988), Apart, Wolfram Language function, https://reference.wolfram.com/language/ref/Apart.html (updated 2007).

Text

Wolfram Research (1988), Apart, Wolfram Language function, https://reference.wolfram.com/language/ref/Apart.html (updated 2007).

CMS

Wolfram Language. 1988. "Apart." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Apart.html.

APA

Wolfram Language. (1988). Apart. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Apart.html

BibTeX

@misc{reference.wolfram_2025_apart, author="Wolfram Research", title="{Apart}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Apart.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_apart, organization={Wolfram Research}, title={Apart}, year={2007}, url={https://reference.wolfram.com/language/ref/Apart.html}, note=[Accessed: 26-March-2025 ]}