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Apart—Wolfram Language Documentation

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• See Also
  • Together
  • ApartSquareFree
  • Cancel
  • PolynomialQuotient
  • Expand
• Related Guides
  • Algebraic Transformations
  • Rational Functions
  • Polynomial Factoring & Decomposition
  • Formula Manipulation
• Tech Notes
  • Putting Expressions into Different Forms
  • Structural Operations on Rational Expressions

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.

• 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.

Decompose into partial fractions:

Compute the partial fraction representation of a rational function:

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:

Compute the partial fraction representation over the integers modulo :

Compute the partial fraction representation following common trigonometric identities:

Partial fraction decomposition over the rationals:

Partial fraction decomposition over the integers modulo 2:

Partial fraction decompositions of trigonometric expressions:

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:

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