Partial fraction
Encyclopedia
In algebra
, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree
of either the numerator or the denominator of a rational function
(also known as a rational algebraic fraction).
In symbols, one can use partial fraction expansion to change a rational function in the form
where ƒ and g are polynomials, into a function of the form
where gj (x) are polynomials that are factors of g(x), and are in general of lower degree.
Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, that produces a single rational function with a numerator and denominator usually of high degree.
The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where:
The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it. Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition. See e.g. partial fractions in integration
for an account of the use of the partial fractions in finding antiderivatives.
Just which polynomials are irreducible depends on which field
of scalars
one adopts. Thus if one allows only real number
s, then irreducible polynomials are of degree either 1 or 2. If complex number
s are allowed, only 1st-degree polynomials can be irreducible. If one allows only rational number
s, or a finite field
, then some higher-degree polynomials are irreducible.
Assume a rational function R(x) = ƒ(x)/g(x) in one indeterminate
x has a denominator that factors as
over a field
K (we can take this to be real number
s, or complex number
s). If P and Q have no common factor, then R may be written as
for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring
over K is a principal ideal domain
, so that
for some polynomials C(x) and D(x) (see Bézout's identity
).
Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomial
s. To take this further, if required, write:
as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm
, polynomial case. The result is the following theorem
:
Therefore when the field K is the complex numbers, we can assume that each pi has degree 1 (by the fundamental theorem of algebra
) the numerators will be constant. When K is the real numbers, some of the pi might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur. This therefore is a case that requires discussion, in the systematic theory of integration (for example in computer algebra).
and 
, where the αi are distinct constants and deg P < n, partial fractions are generally obtained by supposing that
and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients
.)
This approach does not account for several other cases, but can be modified accordingly:
Clearing denominators shows that . Expanding and equating the coefficients of powers of x gives
Solving for A and B yields A = 13/2 and B = −3/2. Hence,
Let
then according to the uniqueness of Laurent series, aij is the coefficient of the term (x − xi)−1 in the Laurent expansion of gij(x) about the point xi, i.e., its residue
This is given directly by the formula
or in the special case when xi is a simple root,
when
Note that P(x) and Q(x) may or may not be polynomials.
integral calculus to find real-valued antiderivative
s of rational function
s. Partial fraction decomposition of real rational function
s is also used to find their Inverse Laplace transforms. For applications of partial fraction decomposition over the reals, see
s. In other words, suppose there exist real polynomials p(x) and q(x)≠ 0, such that
By removing the leading coefficient of q(x), we may assume without loss of generality
that q(x) is monic
. By the fundamental theorem of algebra
, we can write
where a1,..., am, b1,..., bn, c1,..., cn are real numbers with bi2 - 4ci < 0, and j1,..., jm, k1,..., kn are positive integers. The terms (x - ai) are the linear factors of q(x) which correspond to real roots of q(x), and the terms (xi2 + bix + ci) are the irreducible quadratic factors of q(x) which correspond to pairs of complex
conjugate roots of q(x).
Then the partial fraction decomposition of ƒ(x) is the following:
Here, P(x) is a (possibly zero) polynomial, and the Air, Bir, and Cir are real constants. There are a number of ways the constants can be found.
The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants Air, Bir, and Cir. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which always has a unique solution. This solution can be found using any of the standard methods of linear algebra
.
Here, the denominator splits into two distinct linear factors:
so we have the partial fraction decomposition
Multiplying through by x2 + 2x - 3, we have the polynomial identity
Substituting x = -3 into this equation gives A = -1/4, and substituting x = 1 gives B = 1/4, so that
After long-division, we have
Since (−4)2 − 4(8) = −16 < 0, x2 − 4x + 8 is irreducible, and so
Multiplying through by x3 − 4x2 + 8x, we have the polynomial identity
Taking x = 0, we see that 16 = 8A, so A = 2. Comparing the x2 coefficients, we see that 4 = A + B = 2 + B, so B = 2. Comparing linear coefficients, we see that −8 = −4A + C = −8 + C, so C = 0. Altogether,
The following example illustrates almost all the "tricks" one would need to use short of consulting a computer algebra system
.
After long-division and factoring, we have
The partial fraction decomposition takes the form
Multiplying through by (x − 1)3(x2 + 1)2 we have the polynomial identity
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
of either the numerator or the denominator of a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
(also known as a rational algebraic fraction).
In symbols, one can use partial fraction expansion to change a rational function in the form
where ƒ and g are polynomials, into a function of the form
where gj (x) are polynomials that are factors of g(x), and are in general of lower degree.
Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, that produces a single rational function with a numerator and denominator usually of high degree.
The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where:
- the denominator of each term is a powerExponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of an irreducibleIrreducible polynomialIn mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
(not factorable) polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
and - the numerator is a polynomial of smaller degree than that irreducible polynomial. To decrease the degree of the numerator directly, the Euclidean algorithmEuclidean algorithmIn mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
can be used, but in fact if ƒ already has lower degree than g this isn't helpful.
The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it. Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition. See e.g. partial fractions in integration
Partial fractions in integration
In integral calculus, partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions...
for an account of the use of the partial fractions in finding antiderivatives.
Just which polynomials are irreducible depends on which field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of scalars
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
one adopts. Thus if one allows only real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, then irreducible polynomials are of degree either 1 or 2. If complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s are allowed, only 1st-degree polynomials can be irreducible. If one allows only rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, or a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, then some higher-degree polynomials are irreducible.
Basic principles
The basic principles involved are quite simple; it is the algorithmic aspects that require attention in particular cases. On the other hand, the existence of a decomposition of a certain kind is an assumption in practical cases, and the principles should explain which assumptions are justified.Assume a rational function R(x) = ƒ(x)/g(x) in one indeterminate
Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that does not stand for anything else but itself. In particular it does not designate a constant, or a parameter of the problem, it is not an unknown that could be solved for, it is not a variable designating a...
x has a denominator that factors as
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K (we can take this to be real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s). If P and Q have no common factor, then R may be written as
for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
over K is a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, so that
for some polynomials C(x) and D(x) (see Bézout's identity
Bézout's identity
In number theory, Bézout's identity for two integers a, b is an expressionwhere x and y are integers , such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers...
).
Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
s. To take this further, if required, write:
as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
, polynomial case. The result is the following theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
:
Therefore when the field K is the complex numbers, we can assume that each pi has degree 1 (by the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
) the numerators will be constant. When K is the real numbers, some of the pi might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur. This therefore is a case that requires discussion, in the systematic theory of integration (for example in computer algebra).
Procedure
Given two polynomials and , where the αi are distinct constants and deg P < n, partial fractions are generally obtained by supposing that
and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients
Method of undetermined coefficients
In mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...
.)
This approach does not account for several other cases, but can be modified accordingly:
- If deg P
deg Q, then it is necessary to perform the division
-
-
- via polynomial long divisionPolynomial long divisionIn algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...
or otherwise, and then seek partial fractions for the remainder fraction (which by definition has deg R < deg Q).- If Q(x) contains factors which are irreducible over the given field, then the numerator N(x) of each partial fraction with such a factor F(x) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant. For example, take the following decomposition over R:
-
- Suppose Q(x) = (x − α)rS(x) and S(α) ≠ 0. Then Q(x) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of (x − α). For illustration, take S(x) = 1 to get the following decomposition:
-
Illustration
In an example application of this procedure, can be decomposed in the form
Clearing denominators shows that . Expanding and equating the coefficients of powers of x gives
- 5 = A + B and 3x = −2Bx
Solving for A and B yields A = 13/2 and B = −3/2. Hence,
Residue method
Over the complex numbers, suppose ƒ(x) is a rational proper fraction, and can be decomposed into
Let
then according to the uniqueness of Laurent series, aij is the coefficient of the term (x − xi)−1 in the Laurent expansion of gij(x) about the point xi, i.e., its residue
Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
This is given directly by the formula
or in the special case when xi is a simple root,
when
Note that P(x) and Q(x) may or may not be polynomials.
Over the reals
Partial fractions are used in real-variableReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
integral calculus to find real-valued antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
s of rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s. Partial fraction decomposition of real rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s is also used to find their Inverse Laplace transforms. For applications of partial fraction decomposition over the reals, see
- Partial fractions in integrationPartial fractions in integrationIn integral calculus, partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions...
- Partial fractions in Laplace transforms
General result
Let ƒ(x) be any rational function over the real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. In other words, suppose there exist real polynomials p(x) and q(x)≠ 0, such that
By removing the leading coefficient of q(x), we may assume without loss of generality
Without loss of generality
Without loss of generality is a frequently used expression in mathematics...
that q(x) is monic
Monic
In mathematics, monic can refer to*monic morphism - a special kind of morphism in category theory.*monic polynomial - a polynomial whose leading coefficient is one.In linguistics, monic can refer to*Monic languages...
. By the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, we can write
where a1,..., am, b1,..., bn, c1,..., cn are real numbers with bi2 - 4ci < 0, and j1,..., jm, k1,..., kn are positive integers. The terms (x - ai) are the linear factors of q(x) which correspond to real roots of q(x), and the terms (xi2 + bix + ci) are the irreducible quadratic factors of q(x) which correspond to pairs of complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
conjugate roots of q(x).
Then the partial fraction decomposition of ƒ(x) is the following:
Here, P(x) is a (possibly zero) polynomial, and the Air, Bir, and Cir are real constants. There are a number of ways the constants can be found.
The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants Air, Bir, and Cir. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which always has a unique solution. This solution can be found using any of the standard methods of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
.
Example 1
Here, the denominator splits into two distinct linear factors:
so we have the partial fraction decomposition
Multiplying through by x2 + 2x - 3, we have the polynomial identity
Substituting x = -3 into this equation gives A = -1/4, and substituting x = 1 gives B = 1/4, so that
Example 2
After long-division, we have
Since (−4)2 − 4(8) = −16 < 0, x2 − 4x + 8 is irreducible, and so
Multiplying through by x3 − 4x2 + 8x, we have the polynomial identity
Taking x = 0, we see that 16 = 8A, so A = 2. Comparing the x2 coefficients, we see that 4 = A + B = 2 + B, so B = 2. Comparing linear coefficients, we see that −8 = −4A + C = −8 + C, so C = 0. Altogether,
The following example illustrates almost all the "tricks" one would need to use short of consulting a computer algebra system
Computer algebra system
A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
.
Example 3
After long-division and factoring, we have
The partial fraction decomposition takes the form
Multiplying through by (x − 1)3(x2 + 1)2 we have the polynomial identity
-
Taking x = 1 gives 4 = 4C, so C = 1. Similarly, taking x = iComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
gives 2 + 2i = (Fi + G)(2 + 2i), so Fi + G = 1, so F = 0 and G = 1 by equating real and imaginaryComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
parts. With C = G = 1 and F = 0, taking x = 0 we get A - B + 1 - E - 1 = 0, thus E = A - B.
We now have the identity
-
Expanding and sorting by exponents of x we get
-
We can now compare the coefficients and see that
-
with A = 2 - D we get A = D = 1 and so B = 0, furthermore is C = 1, E = A - B = 1, F = 0 and G = 1.
The partial fraction decomposition of ƒ(x) is thus
The role of the Taylor polynomial
The partial fraction decomposition of a rational function can be related to Taylor's theoremTaylor's theoremIn calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...
as follows. Let
be real or complex polynomials; assume that
that
and that
uyu
Define also
Then we have
if, and only if, for each
the polynomial 
is the Taylor polynomial of 
of order 
at the point 
:
Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients.
Sketch of the proof: The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion
, as 
so
is the Taylor polynomial of 
, because of the unicity of the polynomial expansion of order 
, and by assumption 
.
Conversely, if the
are the Taylor polynomials, the above expansions at each 
hold, therefore we also have
, as 
which implies that the polynomial
is divisible by 
For
also 
is divisible by 
, so we have in turn that 
is divisible by 
. Since 
we then have
, and we find the partial fraction decomposition dividing by 
.
Fractions of integers
The idea of partial fractions can be generalized to other ringsRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
,
say the ring of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s where prime numbers take the role of irreducible denominators.
E.g., it is:
External links
- http://cajael.com/eng/control/LaplaceT/LaplaceT-1_Example_2_6_OGATA_4editio.php Make partial fraction decompositions with ScilabScilabScilab is an open source, cross-platform numerical computational package and a high-level, numerically oriented programming language. Itcan be used for signal processing, statistical analysis, image enhancement, fluid dynamics simulations, numerical optimization, and modeling and simulation of...
.
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