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En algunas ocasiones se define la transformada con un factor multiplicativo diferente de
1
2
π
{\displaystyle \textstyle {\frac {1}{\sqrt {2\pi }}}}
1
2
π
{\displaystyle \textstyle {\frac {1}{2\pi }}}
Función
Transformada
δ
(
t
)
{\displaystyle \delta (t)\!}
1
{\displaystyle 1\!}
u
(
t
)
{\displaystyle u(t)\!}
Función unitaria de Heaviside )
1
/
2
(
δ
(
f
)
+
1
/
(
i
π
f
)
)
{\displaystyle 1/2(\delta (f)+1/(i\pi f))\!}
sin
(
w
0
t
)
{\displaystyle \sin(w_{0}t)\!}
π
i
[
δ
(
w
−
w
0
)
−
δ
(
w
+
w
0
)
]
{\displaystyle {\frac {\pi }{i}}[\delta (w-w_{0})-\delta (w+w_{0})]\!}
cos
(
w
0
t
)
{\displaystyle \cos(w_{0}t)\!}
π
[
δ
(
w
−
w
0
)
+
δ
(
w
+
w
0
)
]
{\displaystyle \pi [\delta (w-w_{0})+\delta (w+w_{0})]\!}
1
{\displaystyle 1\!}
δ
(
f
)
=
2
π
δ
(
w
)
{\displaystyle \delta (f)=2\pi \delta (w)\!}
e
−
a
t
u
(
t
)
,
R
e
(
a
)
>
0
{\displaystyle e^{-at}u(t),\quad \mathrm {Re} (a)>0\!}
1
a
+
i
w
{\displaystyle {\frac {1}{a+iw}}\!}
e
−
a
|
t
|
,
{\displaystyle e^{-a|t|},\!}
2
a
a
2
+
w
2
{\displaystyle {\frac {2a}{a^{2}+w^{2}}}\!}
t
e
−
a
t
u
(
t
)
,
R
e
(
a
)
>
0
{\displaystyle te^{-at}u(t),\quad \mathrm {Re} (a)>0\!}
1
(
a
+
i
w
)
2
{\displaystyle {\frac {1}{(a+iw)^{2}}}\!}
{
cos
w
0
x
|
x
|
≤
A
0
|
x
|
>
A
{\displaystyle {\begin{cases}\cos w_{0}x&|x|\leq A\\0&|x|>A\end{cases}}}
sin
A
(
w
−
w
0
)
2
π
(
w
−
w
0
)
+
sin
A
(
w
+
w
0
)
2
π
(
w
+
w
0
)
{\displaystyle {\frac {\sin A(w-w_{0})}{2\pi (w-w_{0})}}+{\frac {\sin A(w+w_{0})}{2\pi (w+w_{0})}}}
x
(
t
)
=
{
1
,
si
|
t
|
<
T
1
0
,
si
|
t
|
>
T
1
{\displaystyle x(t)={\begin{cases}1,&{\mbox{si }}|t|<T_{1}\\0,&{\mbox{si }}|t|>T_{1}\end{cases}}\!}
2
T
1
s
i
n
c
(
w
T
1
π
)
=
2
sin
(
w
T
1
)
w
{\displaystyle 2T_{1}\mathrm {sinc} \left({\frac {wT_{1}}{\pi }}\right)=2{\frac {\sin(wT_{1})}{w}}}
x
(
t
)
=
t
r
i
(
t
2
T
1
)
=
{
1
−
|
t
|
T
1
,
si
|
t
|
<
T
1
0
,
si
|
t
|
>
T
1
{\displaystyle x(t)=\mathrm {tri} \left({\frac {t}{2T_{1}}}\right)={\begin{cases}1-{\frac {|t|}{T_{1}}},&{\mbox{si }}|t|<T_{1}\\0,&{\mbox{si }}|t|>T_{1}\end{cases}}\!}
s
i
n
c
2
(
w
T
1
π
)
{\displaystyle \mathrm {sinc} ^{2}\left({\frac {wT_{1}}{\pi }}\right)}
x
(
t
)
=
e
−
t
2
/
a
2
,
I
m
(
a
)
=
0
{\displaystyle x(t)=e^{-t^{2}/a^{2}},\quad \mathrm {Im} (a)=0\!}
a
2
e
−
a
2
w
2
/
4
{\displaystyle {\frac {a}{\sqrt {2}}}e^{-a^{2}w^{2}/4}}
![{\displaystyle {\frac {\pi }{i}}[\delta (w-w_{0})-\delta (w+w_{0})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5bc8aa58f709a5aab0e2de1aef8241e8d22a87)
![{\displaystyle \pi [\delta (w-w_{0})+\delta (w+w_{0})]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c26f2fe8b7abb13ddaca7dd25fe4443225e4a2)
