Page 505 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

492 Introduction to Laplace Transformation Appen. B

Transforms Having Simple Poles
Considering the subsidiary equation
Ajs)
x ( i) =
B(s)
we examine the case where B(s) is factorable in terms of n roots which are
distinct (simple poles).

B { s ) = (s - a,)(5 - flj) • • • (^ - a „ )
The subsidiary equation can then be expanded in the following partial fractions:
Ajs) ^ _ C ^ C, C.
x(,y) + • • • + (B-8)
B{s) s —
To determine the constants we multiply both sides of the preceding
equation by {s - and let s = a,^. Every term on the right will then be zero
except and we arrive at the result

Ck = lim (B-9)
B(s)

Because £ 'Cf^/(s —a^.) = Qe"*', the inverse transform of x(s) becomes
x(t) = 2^ lim (5 - ; e Qkt (B-10)
1
Another expression for the last equation becomes apparent by noting that
B{s) = {s - a,^)B,{s)
B \ s ) = {s - a,)B\{s) + B,{s)
lim B'{s) = B^{a,^)

Because {s — a,^)A{s)/B{s) = A{s)/B^{s), it is evident that

A O - £ (B-11)
/t=i ^
Transforms Having Poles of Higher Order

If in the subsidary equation
x ( i) =
B(s)
a factor in B{s) is repeated m times, we say that jcCi") has an mth-order pole.