The_Cambridge_Handbook_of_Physics_Formulas

2.12 Laplace transforms
55
2.12 Laplace transforms
Laplace transform theorems
Definition a F(s)=L{f(t)} =
Convolution b
∫ ∞
0
f(t)e −st dt (2.514)
{∫ ∞
}
F(s)·G(s)=L f(t−z)g(z)dz (2.515)
0
= L{f(t)∗g(t)} (2.516)
L{}
F(s)
G(s)
∗
Laplace
transform
L{f(t)}
L{g(t)}
convolution
2
Inverse c f(t)= 1
2πi
∫ γ+i∞
γ−i∞
e st F(s)ds (2.517)
= ∑ residues (for t>0) (2.518)
γ
constant
Transform of
derivative
Derivative of
transform
{ d n }
f(t)
L
dt n
∑n−1
= s n L{f(t)}−
r=0
s n−r−1 dr f(t)
∣
∣∣t=0
dt r
(2.519)
d n F(s)
ds n = L{(−t) n f(t)} (2.520)
n integer > 0
Substitution F(s−a)=L{e at f(t)} (2.521) a constant
Translation
e −as F(s)=L{u(t−a)f(t−a)} (2.522)
{
0 (t0)
u(t)
unit step
function
a If |e −s 0t f(t)| is finite for sufficiently large t, the Laplace transform exists for s>s 0 .
b Also known as the “faltung (or folding) theorem.”
c Also known as the “Bromwich integral.” γ is chosen so that the singularities in F(s) are left of the integral line.