0195136047.pdf

3.3 LAPLACE TRANSFORM 147

Multiple Poles

Let us consider thatF 1 (s) has all simple poles except, say, ats=p 1 which has a multiplicitym.
Then one can write

F 1 (s)=

N 2 (s)

(s−p 1 )m(s−p 2 )···(s−pn)

(3.3.22)

The partial fraction expansion ofF 1 (s) is given by

F 1 (s)=

K 11

(s−p 1 )m

+

K 12

(s−p 1 )m−^1

+···+

K 1 m

(s−p 1 )

+

K 2

(s−p 2 )

+···+

Kn

(s−pn)

(3.3.23)

When a multiple root is involved, there will be as many coefficients associated with the multiple
root as the order of multiplicity. For each simple polepkwe have just one coefficientKk,
as before.
For simple poles one can proceed as discussed earlier and apply Equation (3.3.13) to calculate
the residuesKk. To evaluateK 11 ,K 12 ,...,K 1 mwe multiply both sides of Equation (3.3.23) by
(s−p 1 )mto obtain

(s−p 1 )mF 1 (s)=K 11 +(s−p 1 )K 12 +···

+(s−p 1 )m−^1 K 1 m+(s−p 1 )m

×

(

K 2

s−p 2

+···+

Kn

s−pn

)

(3.3.24)

The coefficientK 11 may now be evaluated as

K 11 =lim

s→p 1

[

(s−p 1 )mF 1 (s)

]

(3.3.25)

Next we differentiate Equation (3.3.24) with respect tosand lets→p 1 in order to evaluateK 12 ,

K 12 =lim

s→p 1

{

d

ds

[

(s−p 1 )mF 1 (s)

]

}

(3.3.26)

The differentiation process can be continued to find thekth coefficient:

K 1 k=lim

s→p 1

{

1

(k− 1 )!

dk−^1

dsk−^1

[

(s−p 1 )mF 1 (s)

]

}

, fork= 1 , 2 ,···,m (3.3.27)

Note thatK 2 ,...,Knterms play no role in determining the coefficientsK 11 ,K 12 ,...,K 1 mbecause
of the multiplying factor (s−p 1 )min Equation (3.3.24).
The alternate representation discussed for the case of complex poles may also be extended
for multiple poles by combining the terms in Equation (3.3.23) corresponding to the multiple root.
In an expansion of a quotient of polynomials by partial fractions, it may, in general, be necessary
to use a combination of the rules given.
The denominator ofF 1 (s) may not be known in the factored form in some cases, and it will
then become necessary to find the roots of the denominator polynomial. If the order is higher
than a quadratic and simple inspection (or some engineering approximation) does not help, one
may have to take recourse to the computer. Computer programs for finding roots of a polynomial
equation (using such methods as Newton–Raphson) are available these days in most computer
libraries.