Partial fraction decomposition

The partial fraction decomposition or partial fraction expansion is a standardized representation of rational functions . It is used in mathematics to make calculations easier with such functions. In particular, it is used in the integration of the rational functions.

This is based on the fact that every rational function is the sum of a polynomial function and fractions of the form

${\displaystyle {\frac {a}{(x-x_{i})^{j}}}}$

can be represented. They are the poles of the function. ${\displaystyle x_{i}}$

If the poles are assumed to be known, then the determination of the numerators is the actual task of the partial fraction decomposition. ${\displaystyle a}$

In real-valued functions, the poles and consequently the numbers do not necessarily have to be real, because the real numbers are not algebraically closed . You can avoid calculating with complex numbers , because with every complex zero the conjugate complex number is also a zero. ${\displaystyle x_{i}}$ ${\displaystyle a}$ ${\displaystyle z_{i}}$ ${\displaystyle {\overline {z_{i}}}}$

Instead and you then used a term , with one real quadratic polynomial is and and are real. ${\displaystyle {\tfrac {a_{1}}{(x-z_{i})^{j}}}}$ ${\displaystyle {\tfrac {a_{2}}{(x-{\overline {z_{i}}})^{j}}}}$ ${\displaystyle {\tfrac {b+cx}{(x^{2}+px+q)^{j}}}}$ ${\displaystyle {x^{2}+px+q}=(x-z_{i})\cdot (x-{\overline {z_{i}}})}$ ${\displaystyle b}$ ${\displaystyle c}$

history

The partial fraction decomposition was developed from 1702 in work on the infinitesimal calculus by Gottfried Wilhelm Leibniz and Johann I Bernoulli . Both scholars use this method to integrate fractional rational functions . Since at that time the fundamental theorem of algebra had not yet been proven - but it was already suspected at the time - Leibniz claimed that there was no partial fraction decomposition for the denominator polynomial . Johann Bernoulli did not agree with this opinion. This example was discussed by various mathematicians in the following years and around 1720 several papers appeared that proved the example to be faulty and the (indefinite) integral ${\displaystyle x^{4}+a^{4}=\left(xx-aa{\sqrt {-1}}\right)\left(xx+aa{\sqrt {-1}}\right)}$

${\displaystyle \int {\frac {\mathrm {d} x}{x^{4}+a^{4}}}}$

correctly calculated.

Procedure

The partial fraction decomposition of a real rational function is determined in several steps: ${\displaystyle R}$

1. One compares the degree of the numerator with that of the denominator of : ${\displaystyle R}$
   • If the count rate is higher or equal to the denominator, so divided to the numerator by the denominator. It obtained the polynomial and possibly a rational residual function , so that: . ${\displaystyle P}$ ${\displaystyle R^{*}={\tfrac {Z^{*}}{N^{*}}}}$ ${\displaystyle R(x)=P(x)+R^{*}(x)}$
     • If the procedure is complete. ${\displaystyle R^{*}\equiv 0}$
     • Otherwise, the counter is of a lesser degree than the denominator . You then only continue to work with the residual function . ${\displaystyle Z^{*}}$ ${\displaystyle R^{*}}$ ${\displaystyle N^{*}}$ ${\displaystyle R^{*}}$
   • If the numerator degree is smaller than the denominator degree, the function can be viewed directly. In order to enable a uniform notation in the following, we use in this case . ${\displaystyle R}$ ${\displaystyle R^{*}:=R}$
3. Then consider the zeros of . A suitable approach is used depending on the type of zeros. ${\displaystyle N^{*}}$
5. The constants , and are then obtained, for example, by comparing coefficients after multiplying the decomposition by the denominator polynomial. ${\displaystyle a_{ij}}$ ${\displaystyle b_{ij}}$ ${\displaystyle c_{ij}}$

The last two steps will now be explained in detail.

approach

It is assumed here that the form is given, where the degree of is smaller than the degree of the denominator polynomial and all zeros of are known. If, as assumed above, the various zeros and their respective degrees are known, the denominator polynomial can be reduced to the following form: ${\displaystyle R^{*}}$ ${\displaystyle R^{*}(x)={\tfrac {Z^{*}(x)}{N^{*}(x)}}}$ ${\displaystyle Z^{*}}$ ${\displaystyle N^{*}}$ ${\displaystyle N^{*}}$ ${\displaystyle n}$ ${\displaystyle x_{i}}$ ${\displaystyle r_{i}}$

${\displaystyle N^{*}(x)=(x-x_{1})^{r_{1}}\cdot (x-x_{2})^{r_{2}}\dotsm (x-x_{n})^{r_{n}}}$

Note that some of the may be non-real . ${\displaystyle x_{i}}$

The approach is now structured as follows:

• For every simple real zero , the approach contains a summand . ${\displaystyle x_{i}}$ ${\displaystyle {\tfrac {a_{i1}}{x-x_{i}}}}$

• The approach contains summands for every -fold real zero . ${\displaystyle r_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle r_{i}}$ ${\displaystyle {\tfrac {a_{i1}}{x-x_{i}}}+{\tfrac {a_{i2}}{(x-x_{i})^{2}}}+\dotsb +{\tfrac {a_{ir_{i}}}{(x-x_{i})^{r_{i}}}}}$

Since is real, the conjugate complex zero belongs to every non-real zero . Let be the quadratic polynomial with zeros and , so . ${\displaystyle R}$ ${\displaystyle z_{i}}$ ${\displaystyle {\overline {z_{i}}}}$ ${\displaystyle x^{2}+p_{i}x+q_{i}}$ ${\displaystyle z_{i}}$ ${\displaystyle {\overline {z_{i}}}}$ ${\displaystyle x^{2}+p_{i}x+q_{i}:=(x-z_{i})(x-{\overline {z_{i}}})}$

• The approach now contains a summand for every simple non-real zero . ${\displaystyle z_{i}}$ ${\displaystyle {\tfrac {b_{i}x+c_{i}}{x^{2}+p_{i}x+q_{i}}}}$

• Correspondingly, the approach contains the terms for every -fold non-real zero (and the associated, also -fold, complex conjugate zero ) . ${\displaystyle s_{i}}$ ${\displaystyle z_{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle {\overline {z_{i}}}}$ ${\displaystyle s_{i}}$ ${\displaystyle {\tfrac {b_{i1}x+c_{i1}}{x^{2}+p_{i}x+q_{i}}}+{\tfrac {b_{i2}x+c_{i2}}{(x^{2}+p_{i}x+q_{i})^{2}}}+\dotsb +{\tfrac {b_{is_{i}}x+c_{is_{i}}}{(x^{2}+p_{i}x+q_{i})^{s_{i}}}}}$

Each approach thus contains exactly unknown coefficients . ${\displaystyle g}$ ${\displaystyle a_{i1},\dotsc ,a_{ir_{i}},b_{i1},\dots ,b_{is_{i}},c_{i1},\dotsc ,c_{is_{i}}}$

Determination of the constants

To determine the constants , and , equation is used with the approach and this equation is multiplied by the denominator polynomial . ${\displaystyle a_{ij}}$ ${\displaystyle b_{ij}}$ ${\displaystyle c_{ij}}$ ${\displaystyle R^{*}}$ ${\displaystyle N^{*}}$

On one side of the equation there is only the numerator polynomial , on the other there is an expression with all unknowns, which is also a polynomial in and can be ordered according to the powers of . A comparison of the coefficients on the left and right then results in a linear system of equations from which the unknown constants can be calculated. Alternatively, up to any different values ​​for can be inserted into this equation, which, like the coefficient comparison, leads to a linear system of equations consisting of equations. It makes sense to insert the previously calculated (real) zeros, which immediately delivers a coefficient value. ${\displaystyle Z^{*}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle g}$ ${\displaystyle x}$ ${\displaystyle g}$

These two possibilities can also be combined.

Examples

Simple poles

The rational function is given

${\displaystyle R(x)={\frac {x}{x^{2}-1}}}$.

There are two simple poles and . So the approach is ${\displaystyle x_{1}=1}$ ${\displaystyle x_{2}=-1}$

${\displaystyle {\frac {x}{x^{2}-1}}={\frac {a_{1}}{x-1}}+{\frac {a_{2}}{x+1}}}$,

where and are unknown constants yet to be determined. Multiplying both sides of the equation by gives you ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle (x^{2}-1)=(x+1)(x-1)}$

${\displaystyle x=a_{1}(x+1)+a_{2}(x-1)}$.

If you sort the right side according to links with and links without , the result is ${\displaystyle x}$ ${\displaystyle x}$

${\displaystyle x=(a_{1}+a_{2})x+(a_{1}-a_{2})}$.

Comparing the coefficients: The coefficient of is one: and the absolute zero element: . From this it can be calculated: The partial fraction we are looking for is therefore ${\displaystyle x}$ ${\displaystyle a_{1}+a_{2}=1}$ ${\displaystyle a_{1}-a_{2}=0}$ ${\displaystyle a_{1}=a_{2}={\tfrac {1}{2}}.}$

${\displaystyle {\frac {x}{x^{2}-1}}={\frac {1}{2}}\cdot {\frac {1}{x-1}}+{\frac {1}{2}}\cdot {\frac {1}{x+1}}.}$

Double poles

The rational function is given

${\displaystyle R(x)={\frac {x^{2}}{x^{2}-2x+1}}}$.

With polynomial division and factorization of the denominator follows

${\displaystyle R(x)={\frac {x^{2}}{x^{2}-2x+1}}=1+{\frac {2\;x-1}{(x-1)^{2}}}}$.

The only, but double zero of the denominator is . Approach: ${\displaystyle x_{0}=1}$

${\displaystyle {\frac {2\;x-1}{(x-1)^{2}}}={\frac {a_{1}}{x-1}}+{\frac {a_{2}}{(x-1)^{2}}}\quad |\cdot (x-1)^{2}}$

${\displaystyle 2\;x-1=a_{1}(x-1)+a_{2}}$

${\displaystyle 2\;x-1=a_{1}x-a_{1}+a_{2}}$

Comparison of coefficients:

${\displaystyle a_{1}=2}$

${\displaystyle -a_{1}+a_{2}=-1}$

Solution:

${\displaystyle a_{1}=2,\quad a_{2}=1}$,

so we get the partial fraction decomposition

${\displaystyle {\frac {x^{2}}{x^{2}-2x+1}}=1+{\frac {2}{x-1}}+{\frac {1}{(x-1)^{2}}}}$.

Complex poles

The rational function is given

${\displaystyle R(x)={\frac {5x^{2}+2x+1}{x^{3}+x}}}$.

The denominator here has the real zero , the complex zero and their complex conjugate . The quadratic polynomial with zeros and is ${\displaystyle x_{1}=0}$ ${\displaystyle x_{2}=z_{1}=\mathrm {i} }$ ${\displaystyle x_{3}={\overline {z_{1}}}=-\mathrm {i} }$ ${\displaystyle z_{1}}$ ${\displaystyle {\overline {z_{1}}}}$ ${\displaystyle (x-z_{1})(x-{\overline {z_{1}}})=(x-\mathrm {i} )(x+\mathrm {i} )=x^{2}+1}$

Approach:

${\displaystyle {\frac {5x^{2}+2x+1}{x^{3}+x}}={\frac {a_{1}}{x}}+{\frac {b_{1}x+c_{1}}{x^{2}+1}}}$

${\displaystyle 5x^{2}+2x+1=a_{1}x^{2}+a_{1}+b_{1}x^{2}+c_{1}x}$

${\displaystyle 5x^{2}+2x+1=(a_{1}+b_{1})x^{2}+c_{1}x+a_{1}}$

Comparison of coefficients:

${\displaystyle {\begin{aligned}a_{1}+b_{1}&=5\\c_{1}&=2\\a_{1}&=1\end{aligned}}}$

Solution:

${\displaystyle a_{1}=1,\quad b_{1}=4,\quad c_{1}=2}$,

Partial fraction decomposition:

${\displaystyle {\frac {5x^{2}+2x+1}{x^{3}+x}}={\frac {1}{x}}+{\frac {4x+2}{x^{2}+1}}}$.

The main theorem about partial fraction decomposition

Real-valued functions

Every rational function with the different real poles of the order and the complex poles of the order that are different apart from conjugation has a clearly defined representation ${\displaystyle R\colon D\subset \mathbb {R} \to \mathbb {R} }$ ${\displaystyle m}$ ${\displaystyle x_{i}}$ ${\displaystyle r_{i}}$ ${\displaystyle n}$ ${\displaystyle z_{i}}$ ${\displaystyle s_{i}}$

${\displaystyle R(x)=P(x)+\sum _{i=1}^{m}\sum _{j=1}^{r_{i}}{\frac {a_{ij}}{(x-x_{i})^{j}}}+\sum _{i=1}^{n}\sum _{j=1}^{s_{i}}{\frac {b_{ij}x+c_{ij}}{(x-z_{i})^{j}(x-{\overline {z_{i}}})^{j}}}}$

with a polynomial function and real constants , and . This is called the partial fraction decomposition (abbreviated PBZ ) of . ${\displaystyle P}$ ${\displaystyle a_{ij}}$ ${\displaystyle b_{ij}}$ ${\displaystyle c_{ij}}$ ${\displaystyle R}$

The fractions are called partial or partial fractions of the 1st kind , the fractions partial or partial fractions of the 2nd kind . ${\displaystyle {\tfrac {a_{ij}}{(x-x_{i})^{j}}}}$ ${\displaystyle {\tfrac {b_{ij}x+c_{ij}}{(x-z_{i})^{j}(x-{\overline {z}}_{i})^{j}}}}$

Complex functions

Every rational function with the different poles of the order has a clearly defined representation ${\displaystyle R\colon D\subset \mathbb {C} \to \mathbb {C} }$ ${\displaystyle n}$ ${\displaystyle z_{i}}$ ${\displaystyle r_{i}}$

${\displaystyle R(z)=P(z)+\sum _{i=1}^{n}\sum _{j=1}^{r_{i}}{\frac {a_{ij}}{(z-z_{i})^{j}}}}$

with a polynomial function and complex constants . ${\displaystyle P}$ ${\displaystyle a_{ij}}$

This theorem can be generalized for polynomials over any other algebraically closed skew body .

Applications

The partial fraction decomposition is used, among other things, to integrate rational functions. Since the integrals of all partial fractions are known, integration is always possible if the poles of the function under consideration can be given.

Furthermore, the partial fraction decomposition is used in the Laplace and the z transformation . The transforms of the individual partial fractions can be looked up in tables. This saves an analytical calculation if the term to be transformed can be broken down into corresponding summands.

Integration of the partial fractions

When finding the antiderivatives of partial fractions, six cases can be distinguished, depending on whether the numerator degree is 0 or 1, whether the poles, i.e. the zeros of the denominator, are real or non-real, and whether they are single or multiple.

Partial fractions with real poles

For partial fractions with real poles there are two cases, since the numerator can only have degree 0.

This results in real and simple poles

${\displaystyle (1)\qquad \int {\frac {A}{x-a}}dx=A\cdot \ln |x-a|+c}$

and for real and multiple poles ( ) ${\displaystyle n\neq 1}$

${\displaystyle (2)\qquad \int {\frac {A}{{(x-a)}^{n}}}dx={\frac {-A\cdot (x-a)^{-(n-1)}}{n-1}}+c}$.

Partial fractures with complex poles

There are four cases for partial fractions with complex poles, since the numerator degree can be either 0 or 1.

This results in complex and simple poles and numerator degrees 0

${\displaystyle (3)\qquad \int {\frac {B}{x^{2}+px+q}}dx={\frac {2B}{\sqrt {4q-p^{2}}}}\cdot \arctan \left({\frac {2x+p}{\sqrt {4q-p^{2}}}}\right)+c}$.

The case with complex and simple poles and numerator degree 1 can be reduced to (3)

${\displaystyle (4)\qquad \int {\frac {Bx+C}{x^{2}+px+q}}dx={\frac {B}{2}}\cdot \ln(x^{2}+px+q)+\left(C-{\frac {pB}{2}}\right)\cdot \int {\frac {1}{x^{2}+px+q}}dx+c}$.

For the two cases with multiple poles, antiderivatives cannot be determined directly, but recursion rules can be found. This results in the case with complex and multiple poles ( ) and numerator degree 0 ${\displaystyle n\neq 0}$

${\displaystyle (5)\qquad \int {\frac {B}{(x^{2}+px+q)^{n+1}}}dx={\frac {B}{(4q-p^{2})\cdot n}}\cdot {\frac {2x+p}{(x^{2}+px+q)^{n}}}+{\frac {2}{4q-p^{2}}}\cdot {\frac {2n-1}{n}}\cdot \int {\frac {B}{(x^{2}+px+q)^{n}}}dx+c}$.

The case with complex and multiple poles and numerator degree 1 can be reduced to (5) ( ) ${\displaystyle n\neq 1}$

${\displaystyle (6)\qquad \int {\frac {Bx+C}{(x^{2}+px+q)^{n}}}dx=-{\frac {B}{2(n-1)}}\cdot {\frac {1}{{(x^{2}+px+q)}^{n-1}}}+\left(C-{\frac {pB}{2}}\right)\cdot \int {\frac {1}{(x^{2}+px+q)^{n}}}dx+c}$.

Laurent series development

If a Laurent series expansion of the function is known for each pole , the partial fraction decomposition is obtained very simply as the sum of the main parts of these Laurent series. This way is related to the residual calculus .

Generalization to rational function bodies

The partial fraction decomposition can be generalized for a body to the rational function body . Denoting the normalized irreducible polynomials in the polynomial ring with , so that rational functions of the form are with independent linear and form with the monomials a - base of - a vector space . ${\displaystyle K}$ ${\displaystyle K(X)}$ ${\displaystyle K[X]}$ ${\displaystyle P}$ ${\displaystyle {\tfrac {X^{i}}{p^{j}}}}$ ${\displaystyle p\in P,j\in \mathbb {N} _{+},0\leq i<\deg(p)}$ ${\displaystyle X^{i},i\in \mathbb {N} }$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K(X)}$

literature

• Student dude mathematics II . Bibliographisches Institut & FA Brockhaus, 2004, ISBN 3-411-04275-3 , pp. 316-317
• Charles D. Miller, Margaret L. Lial, David I. Schneider: Fundamentals of College Algebra . 3. Edition. Scott & Foresman / Little & Brown Higher Education, 1990, ISBN 0-673-38638-4 , pp. 364-370
• LD Kudryavtsev: Undetermined coefficients, method of . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Partial Fraction Decomposition . In: MathWorld (English).
• Günter Scheja, Uwe Storch : Textbook of Algebra. Teubner, Stuttgart 1988, ISBN 3-519-02212-5 .

Web links

• Eric W. Weisstein : Partial Fraction Decomposition . In: MathWorld (English).
• Example of a partial fraction decomposition
• Online calculator and annotated interactive examples (Arndt Brünner)
• Video: Integration through partial fraction decomposition . Jörn Loviscach 2011, made available by the Technical Information Library (TIB), doi : 10.5446 / 9912 .
• Video: partial integration, substitution rule, integration by partial fraction decomposition . Jörn Loviscach 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 9987 .

Individual evidence

1. ^ Heinz-Wilhelm Alten : 4000 years of algebra. History, cultures, people . Springer, Berlin a. a. 2003, ISBN 3-540-43554-9 , pp. 285-286 .
2. ↑ Christoph Bock: Elements of Analysis (PDF; 1.2 MB) Section 8.35
3. ^ Günter Scheja, Uwe Storch : Textbook of Algebra. Teubner, Stuttgart 1988, ISBN 3-519-02212-5 , p. 148