# Dirac's Delta Function And Heaviside's Step Function - HP F2226A - 48GII Graphic Calculator User Manual

Graphing calculator.

Laplace transform of a periodic function of period T:
{ L
(
f
Limit theorem for the initial value: Let F(s) = L{f(t)}, then
f
0
Limit theorem for the final value: Let F(s) = L{f(t)}, then
f

## Dirac's delta function and Heaviside's step function

In the analysis of control systems it is customary to utilize a type of functions
that represent certain physical occurrences such as the sudden activation of a
switch (Heaviside's step function, H(t)) or a sudden, instantaneous, peak in an
input to the system (Dirac's delta function, δ(t)). These belong to a class of
functions known as generalized or symbolic functions [e.g., see Friedman, B.,
1956, Principles and Techniques of Applied Mathematics, Dover Publications
Inc., New York (1990 reprint) ].
The formal definition of Dirac's delta function, δ(x), is δ(x) = 0, for x ≠0, and
Also, if f(x) is a continuous function, then
) (
f
t
L
(
)
F
u
du
s
t
1
T
)}
) (
t
f
t
sT
1
0
e
lim
f
) (
t
lim
[
s
F
t
0
s
lim
f
) (
t
lim
[
s
F
t
s
0
δ
( dx
)
=
1
. 0 .
x
(
f
.
st
.
e
dt
(
s
)].
(
s
)].
)
δ
(
)
=
(
x
x
x
dx
f
x
0
0
Page 16-15
).