Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.4 PARTIAL FRACTIONS

Thus any one of the methods listed above shows that

4 x+2

x^2 +3x+2

=

− 2

x+1

+

6

x+2

.

The best method to use in any particular circumstance will depend on the

complexity, in terms of the degrees of the polynomials and the multiplicities of

the roots of the denominator, of the function being considered and, to some

extent, on the individual inclinations of the student; some prefer lengthy but

straightforward solution of simultaneous equations, whilst others feel more at

home carrying through shorter but more abstract calculations in their heads.

1.4.1 Complications and special cases

Having established the basic method for partial fractions, we now show, through

further worked examples, how some complications are dealt with by extensions

to the procedure. These extensions are introduced one at a time, but of course in

any practical application more than one may be involved.

The degree of the numerator is greater than or equal to that of the denominator

Although we have not specifically mentioned the fact, it will be apparent from

trying to apply method (i) of the previous subsection to such a case, that if the

degree of the numerator (m) is not less than that of the denominator (n) then the

ratio of two polynomials cannot be expressed in partial fractions.

To get round this difficulty it is necessary to start by dividing the denominator

h(x) into the numeratorg(x) to obtain a further polynomial, which we will denote

bys(x), together with a functiont(x)thatisa ratio of two polynomials for which

the degree of the numerator is less than that of the denominator. The function

t(x)cantherefore be expanded in partial fractions. As a formula,

f(x)=

g(x)

h(x)

=s(x)+t(x)≡s(x)+

r(x)

h(x)

. (1.45)

It is apparent that the polynomialr(x)istheremainderobtained wheng(x)is

divided byh(x), and, in general, will be a polynomial of degreen−1. It is also

clear that the polynomials(x) will be of degreem−n. Again, the actual division

process can be set out as an algebraic long division sum but is probably more

easily handled by writing (1.45) in the form

g(x)=s(x)h(x)+r(x) (1.46)

or, more explicitly, as

g(x)=(sm−nxm−n+sm−n− 1 xm−n−^1 +···+s 0 )h(x)+(rn− 1 xn−^1 +rn− 2 xn−^2 +···+r 0 )

(1.47)

and then equating coefficients.