Partial Fractions

 

Introduction

Consider the following fractional algebraic function h(x), in which the degree of the numerator f(x) is lower than the degree of the denominator g(x):

 

Partial fraction is the procedure of converting h(x) in a form of summations of fractions: 

 

In the case that the degree of the numerator f(x) is greater than the degree of the denominator g(x), first we divide f(x) by g(x) and then we calculate the partial fractions.

 

The Partial Fractions

The roots of the denominator could be single, repeated or a mixture of single and repeated roots.

 

1) In the case of single roots, the partial fractions will be:

 

the denominator has n single roots and k are constants.

If we won’t use complex numbers, we might have in the denominator a second order or higher order polynomial. The partial fractions will be different, as shown for the second order case:

 

Generally the degree of the numerator of the partial fractions must be in degree, one less than the denominator. So, if the degree of the denominator of a partial fraction is one, then its numerator is a constant, if the degree of the denominator of a partial fraction is two, then its numerator is of degree one, etc.

 

2) In the case of repeated roots, the partial fractions will be:

 

3) In the case of mixed roots:

 

 

*Note that the coefficients for the single roots begin with k1 while the coefficients for the repeated roots begin with b0. As it will be shown in the Heaviside method later, this is done because it makes calculations easier. Of course someone could use his own coefficients.

 

Calculating the Coefficients

There are two main ways of calculating the coefficients of the partial fractions. The first is equating the coefficients by the initial numerator of the fraction. The second is using the Heaviside cover up method equations.

 

1) Equating Coefficients

This method will be shown using an example. We will convert the denominator of the fraction in product form. Then we will write the partial fractions and, by equating the partial fractions with the initial fraction, we will calculate the coefficients. A mixed root denominator will be used for this example.

 

The initial fraction:

 

The denominator in product form (*the calculations at the end of the page):

 

The partial fractions:

 

Equating the initial fraction with the partial fractions:

 

Now, doing some math:

 

To calculate the coefficients, we equate the numerators:

 

We end up with the equation system:

 

Solving the equations system we have the coefficients:

 

And finally the result is:

 

 

2) Heaviside Cover Up Method

Heaviside method is a very practical way of calculating the coefficients of partial fractions. It supposes that the denominator is converted into product of non reducible polynomials. The partial fractions are produced as shown above and the coefficients are calculated by the following equations:

Coefficients for single roots:

where i is the number for the ith single root and ith coefficient.

 

Coefficients for repeated roots:

where a1 is the repeated root,  i is the ith coefficient for the repeated root and m is the order of the repeated root. (**for more than one repeated root, see the note at the end of the page)

 

As numerical example, a fraction with a mixed root denominator will be used.

 

The initial fraction:

 

The denominator in product form (the calculations at the end of the page):

 

The partial fractions:

 

The coefficients are:

 

And finally the result is:

 

This method is much faster than the Equating Coefficients method.

 

*Notes: The conversion of the denominator into product form.

 

 

**Notes: In the case that we have more than one repeated roots, we develop each root as shown above and the methods of calculating the coefficients apply in the same way: