Formulas for Chapter 4
Table of Laplace Transforms
Z
f (t) for t ≥ 0
∞
F (s) = L(f ) =
e−st f (t) dt
0
1
s
1
s−a
1
eat
tn
ta
sin bt
cos bt
n!
sn+1
(n = 0, 1, . . .)
Γ(a + 1)
(a > 0)
sa+1
b
2
s + b2
s
2
s + b2
f 0 (t)
sF (s) − f (0)
f 00 (t)
s2 F (s) − sf (0) − f 0 (0)
f (n) (t)
sn F (s) − s(n−1) f (0) − s(n−2) f 0 (0) − · · · − f (n−1) (0)
tn f (t)
(−1)n
dn F
(s)
dsn
eat f (t)


 0 t≤a
u(t − a) =

 1 t>a
F (s − a)
u(t − a)f (t − a)
e−as F (s)
u(t − a)g(t)
e−as L(g(t + a))
δ(t − a)
Z t
(f ∗ g)(t) =
f (t − τ )g(τ ) dτ
e−as
0
If f (t + T ) = f (t) for all t
Z t
f (τ ) dτ
0
e−as
s
L(f ∗ g) = L(f )L(g)
R
T −sτ
e
f
(τ
)
dτ
0
L(f ) =
1 − e−T s
1
F (s)
s
Partial Fractions
These notes are concerned with decomposing rational functions
aM sM + aM −1 sM −1 + · · · + a1 s + a0
P (s)
=
Q(s)
sN + bN −1 sN −1 + · · · + b1 s + b0
Note: we can (without loss of generality assume that the coefficient of sN in the denominator is 1.
I. Degree P (s) > Degree Q(s) In this case first carry out long division to obtain
P (s)
P2 (s)
= P1 (s) +
Q(s)
Q(s)
where Degree(P2 ) < Degree(Q).
II. Nonrepeated factors
If Q(s) = (s − r1 )(s − r2 ) · · · (s − rn ) and ri 6= rj for i 6= j
A1
A2
An
P (s)
=
+
+ ··· +
Q(s)
(s − r1 ) (s − r2 )
(s − rn )
III. Repeated Linear Factors
If Q(s) contains a factor of the form (s − r)m then you must have the following terms
A2
Am
A1
+
+
·
·
·
+
(s − r) (s − r)2
(s − r)m
IV. A Nonrepeated Quadratic Factor
If Q(s) contains a factor of the form (s2 − 2αs + α2 + β 2 ) = (s − α)2 + β 2 then you
must have the following term
A1 s + B1
2
(s − 2αs + α2 + β 2 )
V. Repeated Quadratic Factors
If Q(s) contains a factor of the form (s2 − 2αs + α2 + β 2 )m then you must have the
following terms
A1 s + B1
A2 s + B2
Am s + Bm
+ 2
+ ··· + 2
2
2
2
2
2
2
(s − 2αs + α + β ) (s − 2αs + α + β )
(s − 2αs + α2 + β 2 )m