Lecture Notes in Differential Equations - Bruce E. Shapiro

349
Summary of Translation Formulas
L[U(t − a)] = e−as
s
(32.155)
L[f(t)U(t − a)] = e −sa L[f(t + a)] (32.156)
L[f(t − a)U(t − a)] = e −as F (s) (32.157)
L [ e at f(t) ] = F (s − a) (32.158)
L −1 [F (s − a)] = e at f(t) (32.159)
Convolution
Definition 32.12 (Convolution). Let f(t) and g(t) be integrable functions
on (0, ∞). Then the convolution of f and g is defined by
(f ∗ g)(t) =
∫ t
Som Useful Properties of the Convolution
1. f ∗ g = g ∗ f (commutative)
2. f ∗ (g + h) = f ∗ g + f ∗ h (distributive)
3. f ∗ (g ∗ h) = (f ∗ g) ∗ h (associative)
4. f ∗ 0 = 0 ∗ f = 0 (convolution with zero is zero)
Example 32.15. Find sin t ∗ cos t
sin t ∗ cos t =
=
∫ t
0
∫ t
0
= cos t
0
f(u)g(t − u)dt (32.160)
sin x cos(t − x)dx (32.161)
sin x(cos t cos x + sin t sin x)dx (32.162)
∫ t
0
sin x cos xdx + sin t
∫ t
= cos t 1 t ( )∣ 2 sin2 x
x sin 2x ∣∣∣
t
∣ + sin t −
0
2 4
0
= 1 ( )
t
2 cos t sin 2t
sin2 t + sin t −
2 4
0
sin 2 xdx (32.163)
(32.164)
(32.165)