HP 49g+, Руководство пользователя

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∫
∞
⋅ ⋅ ⋅ ⋅ = =
0
) sin( ) (
2
) ( )} ( {
dt t t f F t f
ω
π
ω
s
F
Inverse sine transform
∫
∞
−
⋅ ⋅ ⋅ = =
0
1
) sin( ) ( ) ( )} ( {
dt t F t f F
s
ω ω ω F
Fourier cosine transform
∫
∞
⋅ ⋅ ⋅ ⋅ = =
0
) cos( ) (
2
) ( )} ( {
dt t t f F t f
ω
π
ω
c
F
Inverse cosine transform
∫
∞
−
⋅ ⋅ ⋅ = =
0
1
) cos( ) ( ) ( )} ( {
dt t F t f F
c
ω ω ω F
Fourier transform (proper)
∫
∞
∞ −
−
⋅ ⋅ ⋅ = =
dt e t f F t f
t i ω
π
ω ) (
2
1
) ( )} ( { F
Inverse Fourier transform (proper)
∫
∞
∞ −
− −
⋅ ⋅ = =
dt e F t f F
t i ω
ω ω ) ( ) ( )} ( {
1
F
Example 1 – Determine the Fourier transform of the function f(t) = exp(- t), for t
>0, and f(t) = 0, for t<0.
The continuous spectrum, F(
ω
), is calculated with the integral:
∫ ∫
+ −
∞
∞ →
+ − =
ε
ω
ε
ω
π π
0
) 1 (
0
) 1 (
2
1
lim
2
1
dt e dt e
t i t i
.
1
1
2
1
1
) ) 1 ( exp( 1
2
1
lim
ω π ω
ε ω
π
ε
i i
i
+
⋅ =




+
+ − −
=
∞ →
This result can be rationalized by multiplying numerator and denominator by
the conjugate of the denominator, namely, 1-i
ω
. The result is now: