9.61 PartialFraction

A partial fraction is a decomposition of a quotient into a sum of quotients where the denominators of the summands are powers of primes.Most people first encounter partial fractions when they are learning integral calculus. For a technical discussion of partial fractions, see, for example, Lang’s Algebra. For example, the rational number 1/6 is decomposed into 1/2-1/3. You can compute partial fractions of quotients of objects from domains belonging to the category EuclideanDomain. For example, Integer, Complex Integer, and UnivariatePolynomial(x, Fraction Integer) all belong to EuclideanDomain. In the examples following, we demonstrate how to decompose quotients of each of these kinds of object into partial fractions. Issue the system command )show PartialFraction to display the full list of operations defined by PartialFraction.

It is necessary that we know how to factor the denominator when we want to compute a partial fraction. Although the interpreter can often do this automatically, it may be necessary for you to include a call to factor. In these examples, it is not necessary to factor the denominators explicitly.

The main operation for computing partial fractions is called partialFractionpartialFractionPartialFraction and we use this to compute a decomposition of 1 / 10!. The first argument to partialFractionpartialFractionPartialFraction is the numerator of the quotient and the second argument is the factored denominator.

partialFraction(1,factorial 10)

\[\]

15928-2334-1252+17

_{Type: PartialFraction Integer}

Since the denominators are powers of primes, it may be possible to expand the numerators further with respect to those primes. Use the operation padicFractionpadicFractionPartialFraction to do this.

f := padicFraction(%)

\[\]

12+124+125+126+127+128-232-133-234-25-252+17

_{Type: PartialFraction Integer}

The operation compactFractioncompactFractionPartialFraction returns an expanded fraction into the usual form. The compacted version is used internally for computational efficiency.

compactFraction(f)

\[\]

15928-2334-1252+17

_{Type: PartialFraction Integer}

You can add, subtract, multiply and divide partial fractions. In addition, you can extract the parts of the decomposition. numberOfFractionalTermsnumberOfFractionalTermsPartialFraction computes the number of terms in the fractional part. This does not include the whole part of the fraction, which you get by calling wholePartwholePartPartialFraction. In this example, the whole part is just 0.

numberOfFractionalTerms(f)

\[\]

12

_{Type: PositiveInteger}

The operation nthFractionalTermnthFractionalTermPartialFraction returns the individual terms in the decomposition. Notice that the object returned is a partial fraction itself. firstNumerfirstNumerPartialFraction and firstDenomfirstDenomPartialFraction extract the numerator and denominator of the first term of the fraction.

nthFractionalTerm(f,3)

\[\]

125

_{Type: PartialFraction Integer}

Given two gaussian integers (see ComplexXmpPage ), you can decompose their quotient into a partial fraction.

partialFraction(1,- 13 + 14 * %i)

\[\]

-11+2i+43+8i

_{Type: PartialFraction Complex Integer}

To convert back to a quotient, simply use a conversion.

% :: Fraction Complex Integer

\[\]

-i14+13i

_{Type: Fraction Complex Integer}

To conclude this section, we compute the decomposition of

              1
-------------------------------
              2       3       4
(x + 1)(x + 2) (x + 3) (x + 4)

The polynomials in this object have type UnivariatePolynomial(x, Fraction Integer).

We use the primeFactorprimeFactorFactored operation (see FactoredXmpPage ) to create the denominator in factored form directly.

u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4])

\[\]

(x+1)(x+2)2(x+3)3(x+4)4

_{Type: Factored UnivariatePolynomial(x,Fraction Integer)}

These are the compact and expanded partial fractions for the quotient.

partialFraction(1,u)

\[\]

1648x+1+14x+716(x+2)2+-178x2-12x-1398(x+3)3+607324x3+10115432x2+3914x+44179324(x+4)4

_{Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)}

padicFraction %

\[\]

1648x+1+14x+2-116(x+2)2-178x+3+34(x+3)2-12(x+3)3+607324x+4+403432(x+4)2+1336(x+4)3+112(x+4)4

_{Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)}

All see FullPartialFractionExpansionXmpPage for examples of factor-free conversion of quotients to full partial fractions.