Kidoguchi, Kenneth 1 22 November, 2019 1 Rule 2 Quadratic Factor Partial Fractions The portion of the partial fraction decomposition of the rational function R(s) corresponding to the irreducible quadratic factor (s – a) 2 of multiplicity n is a sum of n partial fractions having the form where A 1 , A 2 , A 3 , …., A n , B 1 , B 2 , B 3 , …., B n are constants. 1 1 2 2 2 2 2 2 2 2 2 n n n As B As B As B s a b s a b s a b Rule 1 Linear Factor Partial Fractions The portion of the partial fraction decomposition of the rational function R(s) corresponding to the linear factor s – a of multiplicity n is a sum of n partial fractions having the form where A 1 , A 2 , A 3 , …., and A n are constants. 1 2 2 n n A A A s a s a s a §7.3 Translation and Partial Fractions Partial Fraction Decomposition Rules
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Kidoguchi, Kenneth122 November, 2019 1
Rule 2 Quadratic Factor Partial Fractions
The portion of the partial fraction decomposition of the rational function
R(s) corresponding to the irreducible quadratic factor (s – a)2 of
multiplicity n is a sum of n partial fractions having the form
where A1, A2, A3, …., An , B1, B2, B3, …., Bn are constants.
1 1 2 22 22 2 22 2
n nn
A s BAs B A s B
s a b s a b s a b
Rule 1 Linear Factor Partial Fractions
The portion of the partial fraction decomposition of the rational function
R(s) corresponding to the linear factor s – a of multiplicity n is a sum of
n partial fractions having the form
where A1, A2, A3, …., and An are constants.
1 22
nn
AA A
s a s a s a
§7.3 Translation and Partial Fractions
Partial Fraction Decomposition Rules
22 November, 2019 2 Kidoguchi, Kenneth
2
1( )
2 5G s
s s
Find g(t) = L-1{G(s)} if:
§7.3 Translation and Partial Fractions
Inverse Laplace Transform – PFD Example
22 November, 2019 3 Kidoguchi, Kenneth
22 November, 2019 5 Kidoguchi, Kenneth
2 2( 5)s
5 sin( )te t L 2 2sin t
s
L
2 2
2
( 2) 3
s
s
2 cos(3 )te t L 2 2cos 3
3
st
s
L
( ) ( )ate f t F s a L ( ) ( )f t F sL
0
( ) ( )at st ate f t e dt e f t
L( )
0
Proof: ( ) ( ) s a tF s a f t e dt
If F(s) = L{f(t)} exists for s > c, the L{eat f(t)} exists for s > a + c, and
L{eat f(t)} = F(s – a)
Equivalently: L-1{F(s – a)} = eat f(t)
Thus the translation s s – a in the transform corresponds to