Table of Laplace Transforms
L {f (t)} = F (s) =
Z
∞
e−st f (t) dt
0
Note: All f (t) in the table below are assumed zero for t < 0. So if using this table to find the inverse
1
transform, remember to multiply the result by H(t). For example, the inverse transform of s+a
is H(t)e−at .
f (t)
F (s)
f (t)
F (s)
δ(t)
1
1
1 − e−at
a
1
s(s + a)
1
1
s
te−at
1
(s + a)2
t
1
s2
tn e−at
n!
(s + a)n+1
n!
e−at sin(bt)
b
(s + a)2 + b2
e−at cos(bt)
s+a
(s + a)2 + b2
tn
sn+1
1
s+a
e−at
Properties of the Laplace Transform
• Linearity: L {αf (t) + βg(t)} = αF (s) + βG(s).
• Time shift: L {H(t − τ )f (t − τ )} = e−τ s F (s) for τ ≥ 0.
• Derivative: L {f 0 (t)} = sF (s) − f (0).
• Second derivative: L {f 00 (t)} = s2 F (s) − sf (0) − f 0 (0).
• Higher derivatives: L {f (n) (t)} = sn F (s) − sn−1 f (0) − sn−2 f 0 (0) − · · · − f (n−1) (0).
• Integration: L
Z
t
f (τ ) dτ
0
1
= F (s).
s
• Convolution: L (f ∗ g)(t) = L
Z
t
f (τ )g(t − τ ) dτ
0
= F (s)G(s).