4.8 Inverse Transforms of Rational Functions

Let F(s) be a rational function of s. Inverse Laplace transform of F(s).

Example:

i) If denominator of a function has distinct real roots:

F(s) = (s + 1) (s + 3) find f(t)

s (s + 2) (s + 8)

Solution:

F(s) = 3 + 1 + 35

16s 1(s + 2) 48(s + 8)

Thus,

F(t) = 3 + 1 e^-2t 35 e^-8t

16 12 48

ii) Denominator of a function has distinct complex roots:

If F(s) = 4s + 3 find f(t)

s² + 2s + 5

<Solution: F(s) = 4 s + 1 - 1 2

(s + 1)² + 2² 2 (s + 1)² + 2²

Thus,

F(t) = 4e^-t cos(2t) – 1 e^-1 sin(2t)

iii) Denominator of a function has repeated real roots:

F(s) = 3s + 4 find f(t)

(s + 1) (s² + 4s+ 4)

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