The command PF computes the partial fraction expansion of a rational function. By default, if the rational function contains several variables, PF computes the partial fraction expansion along all the variables. More precisely, PF first computes the partial fraction expansion along the first variable (in the usual lexicographic order) and then, if there is a term whose denominator does not depend on the first variable, expands it along the second variable, and so on, until all the terms have been expanded as much as possible.
In general, the final result will depend on the order in which the expansion along the different variables is performed. Because of that, PF takes a list of variables as an optional second argument. This list of variable specifies along which variables the expansion should be done and the order in which it should be computed. As an example, consider the rational function given in Section 2,
Here is the output of PF, first with no optional argument, then with {x} and {y,x} respectively as optional arguments.
In the first case the expansion was done on x first and then on y; in the second case the expansion was done on x alone; and in the third case the expansion was done on y first and then on x. Note in particular that the first and third outputs are different (though equal, of course), and not merely the same fractions in different orders.