F
{
(
)}
f
t
s
Inverse sine transform
−
1
F
{
(
F
s
Fourier cosine transform
F
{
(
)}
f
t
c
Inverse cosine transform
−
1
F
{
(
F
c
Fourier transform (proper)
F
{
(
f
t
Inverse Fourier transform (proper)
−
1
F
{
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:
1
2
π
1
lim
2
π
ε
→
∞
This result can be rationalized by multiplying numerator and denominator by
the conjugate of the denominator, namely, 1-iω. The result is now:
2
∞
(
ω
)
) (
F
f
t
π
0
∞
ω
)}
) (
(
ω
)
f
t
F
0
2
∞
(
ω
)
) (
F
f
t
π
0
∞
ω
)}
=
) (
=
(
ω
)
f
t
F
0
1
∞
)}
=
(
ω
)
=
⋅
F
f
2
π
−
∞
∞
ω
ω
(
)}
) (
(
f
t
F
−
∞
1
1 (
i
ω
)
t
e
dt
=
lim
2
π
0
ε
1
exp(
1 (
i
ω
)
ε
)
1
i
ω
sin(
ω
)
t
dt
sin(
ω
)
t
dt
cos(
ω
)
t
dt
⋅
cos(
ω
⋅
)
⋅
t
dt
−
iω
t
) (
⋅
⋅
t
e
dt
−
iω
t
)
e
dt
ε
1 (
i
ω
)
t
e
dt
0
1
1
.
2
π
1
i
ω
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