Laplace transform
Laplace transform properties
Caption: X(s) = L[x(t)](s), sx : abscissa of convergence,
U (t): unit step function.
Linearity (a, b ∈ R)
L[ax1 (t) + bx2 (t)](s) = aX1 (s) + bX2 (s)
Time Scaling (a > 0)
L[x(at)](s) =
1
X(s/a)
a
s > asx
Time Shift (a ∈ R)
L[x(t − a) U(t − a)](s) = e−as X(s)
s > sx
Frequency Shift (a ∈ R)
L[eat x(t)](s) = X(s − a)
s > a + sx
Multiplication by time (n ∈ N)
L[tn x(t)](s) = (−1)n
First Derivative
L[x′ (t)](s) = sX(s) − x(0+ )
s > max{sx , sx′ }
Second Derivative
L[x′′ (t)](s) = s2 X(s) − sx(0+ ) − x′ (0+ )
s > max{sx , sx′ , sx′′ }
Integration
L
[∫ t
L
0
]
x(t)
(s) =
t
s > max{0, sx }
X(r) dr
s > sx
∫ +∞
s
Convolution
L[x1 (t) ∗ x2 (t)](s) = X1 (s) X2 (s)
T -periodic function (T > 0)
1
L[x(t)](s) =
1 − e−sT
Initial Value Theorem
Final Value Theorem
c 2018 Politecnico di Torino
⃝
∫ T
0
lim x(t) = lim sX(s)
t→0+
s→+∞
lim x(t) = lim sX(s)
t→+∞
s > sx
X(s)
s
x(u) du (s) =
[
Integral of transform
]
dn
X(s)
dsn
s > max{sx1 , sx2 }
s→0+
e−st x(t) dt
s > max{sx1 , sx2 }
s>0
Inverse Laplace transform properties
Caption: x(t) = L−1 [X(s)](t),
U(t): unit step function.
Linearity (a, b ∈ R)
L−1 [aX1 (s) + bX2 (s)](t) = ax1 (t) + bx2 (t)
Time Scaling (a > 0)
L−1 [X(as)](t) =
Time Shift (a ∈ R)
L−1 [e−as X(s)](t) = x(t − a) U(t − a)
Frequency Shift (a ∈ R)
L−1 [X(s − a)](t) = eat x(t)
Inverse transform of derivative
L−1 [X ′ (s)](t) = −tx(t)
Inverse transform of integral
L−1
Multiplying by frequency
Division by frequency
1
x(t/a)
a
]
[∫ +∞
X(r) dr (t) =
s
x(t)
t
L−1 [sX(s) − x(0+ )](t) = x′ (t)
−1
L
[
]
X(s)
(t) =
s
∫ t
x(u) du
0
c 2018 Politecnico di Torino
⃝
Fundamental Laplace transforms
Function x(t)
Laplace transform X(s)
Domain
1
s
s>0
1
s−a
s>a
U(t)
eat
(a ∈ R)
tn
(n ∈ N)
n!
s>0
sn+1
sin (at)
(a ∈ R)
a
s 2 + a2
s>0
cos (at)
(a ∈ R)
s
s 2 + a2
s>0
sinh (at)
(a ∈ R)
cosh (at)
(a ∈ R)
a
s > |a|
s 2 − a2
s
s > |a|
s 2 − a2
eat sin (bt)
(a, b ∈ R)
b
(s − a)2 + b2
s>a
eat cos (bt)
(a, b ∈ R)
s−a
(s − a)2 + b2
s>a
arctan 1/s
s>0
sin t
t
Distribution x(t)
δ(t) (“Delta function”)
c 2018 Politecnico di Torino
⃝
Laplace transform X(s)
1