Partial Fractions
A partial fraction is a decomposition of a quotient into a sum of quotients
where the denominators of the summand 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 objects into partial fractions.
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
partialFraction and we use this
to compute a decomposition of 1/10!. The first argument top
partialFraction is the numerator
of the quotient and the second argument is the factored denominator.
Since the denominators are powers of primes, it may be possible to expand
the numerators further with respect to those primes. Use the operation
padicFraction to do this.
The operation compactFraction
returns an expanded fraction into the usual form. The compacted version
is used internally for computational efficiency.
You can add, subtract, multiply, and divide partial fractions. In addition,
you can extract the parts of the decomposition.
numberOfFractionalTerms
computes the number of terms in the fractional part. This does not include
the whole part of the fraction, which you get by calling
wholePart. In this example, the whole part
is 0.
The operation
nthFractionalTerm
returns the individual terms in the decomposition. Notice that the object
returned is a partial fraction itself.
firstNumer and
firstDenom extract the numerator and
denominator of the first term of the fraction.
Given two gaussian integers (see Complex),
you can decompose their quotient into a partial fraction.
To convert back to a quotient, simply use the conversion
To conclude this section, we compute the decomposition of
1
-------------------------------
2 3 4
(x + 1)(x + 2) (a + 3) (x + 4)
The polynomials in this object have type
UnivariatePolynomial(x,Fraction Integer).
We use the primeFactor operation
(see Factored) to create the denominator
in factored form directly.
These are the compact and expanded partial fractions for the quotient.
Also see
FullPartialFractionExpansion for examples of factor-free conversion of
quotients to full partial fractions.
Issue the system
command
to display the full list of operations defined by
PartialFraction.
