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The Dirac Delta Function and Impulse Response

and therefore
[ 2 ] t 4
−1
L = .
s 5 12
. . . .

Error Function

Definition 2.1✓ The error function from the theory of probability

is defined as
∫ t
2 2
erf(t) = √ e −x dx.
π 0
. . . . .

Note that ∫
2 ∞ 2
lim erf(t) = √ e −x dx = 1. (2.22)
t→∞ pi 0
The error function is related to Laplace transforms through the problem of finding

( )
1
−1
L √ .
s(s − 1)
It can be proved that
( )
1 1
L √ = √
πt s

t
and also that L(e ) = 1/(s − 1). Then by the convolution theorem,
( ) ∫ t ∫ t
1 1 1 e t e −x
−1 t t−x
L √ = √ ∗ e = √ e dx = √ √ dx.
s(s − 1) πt 0 πx π 0 x
√
Substituting u = x gives

√
( ) ∫ t
1 2e t 2 √
−1 −u t
L √ = √ e du = e erf( t).
s(s − 1) π 0
Applying the first translation theorem with a = −1 yields

√ 1
L(erf( t)) = √ .
s s + 1

3. The Dirac Delta Function and Impulse

Response

In applications, we are often encountered with linear systems, originally at
rest, excited by a sudden large force (such as a large applied voltage to an

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