Find inverse laplace for: f(s)=(1)/((s−2)(s+2)^2)

jatericrimson8b 2022-09-12 Answered
Find inverse laplace for:
f ( s ) = 1 ( s 2 ) ( s + 2 ) 2
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Answers (1)

Baluttor7
Answered 2022-09-13 Author has 17 answers
The inverse Laplace Transform is given by
f ( t ) = 1 2 π i 2 + i 2 + + i 1 ( s 2 ) ( s + 2 ) 2 e s t d s
For t>0, we close the contour in the left-half plane s < 2 + . Note that F ( s ) e s t has a simple pole at s=2 and a second-order pole at s=−2.
The associated residues of F ( s ) e s t are e 2 t 16 at s = 2 and 1 4 t e 2 t 1 16 e 2 t at s = 2
Hence, we find that for t>0
f ( t ) = 1 16 e 2 t 1 16 e 2 t 1 4 t e 2 t
For t<0, we close the contour in the right-half plane. Since F ( s ) e s t is analytic in that plane, f(t)=0 for t<0.

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