Download Delta Function The (Dirac) delta function is defined by δ(x) = { 0 x = 0

Delta Function
The (Dirac) delta function is defined by
δ(x) =
with
R∞
−∞
δ(x) dx = 1 and
R∞
−∞
(
0 x 6= 0
∞ x=0
δ(x − a)f (x) dx = f (a).
KHxL
15
10
5
0
-2
0
-1
1
2
The delta function can be thought of as the limit of a sequence of Gaussians
as the width → 0 and the height → ∞:
(
x2
1
exp −
δ(x) = lim K(x, t) = lim √
t→0
t→0
4t
4πt
)
whereRK(x, t) is the heat equation kernel or fundamental
solution. Note that
R∞
∞
δ(x) dx = 1. The funK(x, t) dx = 1 for any t, in particular −∞
since −∞
damental solution to the heat equation satisfies the initial value problem
∂K
∂ 2K
=
, K(x, t = 0) = δ(x).
∂t
∂x2
The Laplace Transform of the delta function is
L{δ(t)} =
Z
∞
0
e−st δ(t − c) dt = e−cs , c > 0.