1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

564 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

s^3 -4s+l.

• EXAMPLE 12.24 Let Y (s) = 3. Find c-^1 (Y (s)).

s(s- 1)

Solutio n From Equations (12-37) and (12-38) we write

s^3 - 4s - 1 A3 A2 A1 Bi

-----, 3 .-= 3 + 2 +--+-.

s(s-1) (s-1) (s - 1) s - 1 s

We calculate the coefficient B 1 by

Bi = Res (Y, OJ = Jim 53 - 4s + 1 = -1.

•- o (s - 1)3

We find the coefficients Ai, A2, and A3 by using Theorem 12.20. In this case

P (s) s^3 - 4s+l

a=land Q(s) = s ,and we get

A3 = Jim p (s) = lim s3 - 4s + 1 = -2;

•- 1 Q (s) s-1 s

A2 = 111 lim dd QP((s)) = lim (2s - 12) = 1;

. 8 -+l s s s - 1 s

1. d
   2
   P(s) 1. ( 2)

Ai = - 2 hm -d 2 Q ( ) = - 2 hm 2 + 3 = 2.

•- 1 s s •- 1 s

Hence the partial fraction representation is

• 2 1 2 1
  Y(s)= + +-- -
  (s-1)3 (s-1)^2 s - 1 s'

and the inverse is