Math Assignment Help With Inverse Transforms Of Rational Functions
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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