HP 49g+, Руководство пользователя

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Page 16-15
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Laplace transform of a periodic function of period T:
•
Limit theorem for the initial value: Let F(s) = L{f(t)}, then
•
Limit theorem for the final value: Let F(s) = L{f(t)}, then
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
∫
∞
∞ −
= − ). ( ) ( ) (
0 0
x f dx x x x f
δ
∫
∞
=






s
du u F
t
t f
. ) (
) (
L
∫
⋅ ⋅ ⋅
−
=
−
−
T
st
sT
dt e t f
e
t f
0
. ) (
1
1
)} ( { L
)]. ( [ lim ) ( lim
0
0
s F s t f f
s t
⋅ = =
∞ → →
)]. ( [ lim ) ( lim
0
s F s t f f
s t
⋅ = =
→ ∞ →
∞
∫
∞
∞ −
= . 0 . 1 ) (
dx x
δ