Use the Simplify > Expand command984VML or the Expansion FunctionsExpansion_Functions to expand a rational function into partial fraction form with respect to some or all of its variables.


When a rational function is expanded, the following sequence of transformations occurs:

•        If the rational function is improper with respect to the primary expansion variable, it is transformed using polynomial division into an expanded polynomial quotient plus a proper rational functionProper_Rational_Function.

•        The denominator of the proper rational function is factored with respect to the expansion variables.  The amount of factoring is that specified by the Denominator Factoring Type field of the Simplify > Expand command984VML or by the amount argument of the EXPAND or TERMS function.

•        The proper rational function is expanded into a sum of independent super-proper rational functionSuper_proper_Rational_Function terms.  The denominator of each of these terms is one of the factors of the proper rational function's denominator raised to a power not exceeding its power in the denominator.

•        If the numerator of a super-proper term is a sum that contains expansion variables, the denominator of that term is distributed over those terms of the numerator that contain expansion variables.


If the sum of the degrees of the expansion variables in the denominator of a rational function exceeds two, the rational or radical factoring of the denominator can make partial fraction expansion quite slow.  Therefore, expand such rational functions with respect to as few variables as possible.


An alternative to partial fraction expansion is to separately highlight and expand the numerator and denominator of a rational function.  For rational functions with a denominator having expansion variables raised to high degrees, this is usually much quicker than partial fraction expansion.  However, the resulting expression does not clearly indicate the zeros, poles, or residues of the rational function.

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