Dirac's Delta Function And Heaviside's Step Function - HP 49g+ User Manual

•
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
).