Function Repository Resource:

ApartAll

Source Notebook

Compute a partial fraction decomposition over the algebraic closure of the rationals

Contributed by: Nikolay Murzin

ResourceFunction["ApartAll"][expr]

rewrites the rational expression expr as a sum of terms with minimal denominators in an arbitrary number field.

ResourceFunction["ApartAll"][expr,var]

treats all variables other that var as constants.

Details and Options

ResourceFunction["ApartAll"][expr,"NumericFactors"→True] evaluates numerically.

As with Apart, the result of ResourceFunction["ApartAll"] is often called a "partial fraction decomposition". The distinction is that Apart only factors over the rationals, whereas ResourceFunction["ApartAll"] works over the algebraic closure of the rationals.

Examples

Basic Examples (1)

Decompose into partial fractions:

In[1]:=

Out[1]=

Options (1)

NumericFactors (1)

"NumericFactors"→True numerically evaluates factors before the computation:

In[2]:=

Out[2]=

Properties and Relations (2)

ApartAll works like Apart, but factorizes polynomials with Extension→All:

In[3]:=

Out[3]=

Compare outputs of Apart and ApartAll in a case where the polynomial denominator does not factor over the rationals into powers of linear polynomials:

In[4]:=

${ResourceFunction["ApartAll"][1/((x - 1) (x^2 - 6 x + 13) (x^2 + 5))], Apart[1/((x - 1) (x^2 - 6 x + 13) (x^2 + 5))]}$