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Uwe a/latex
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f
(
x
)
=
1
2
π
j
∫
c
+
j
∞
c
−
j
∞
F
(
s
)
e
s
t
d
s
{\displaystyle f(x)={{1 \over {2\pi j}}\int _{c+j\infty }^{c-j\infty }F(s)e^{st}ds}}
F
(
s
)
=
∫
0
∞
f
(
t
)
e
−
s
t
d
t
{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}dt}
δ
(
t
)
{\displaystyle \delta (t)}
1
{\displaystyle 1}
h
(
t
)
{\displaystyle h(t)}
1
s
{\displaystyle 1 \over s}
t
n
,
n
=
1
,
2
,
3
{\displaystyle t^{n},n=1,2,3}
n
!
s
n
+
1
{\displaystyle n! \over s^{n+1}}
t
n
e
−
a
t
{\displaystyle t^{n}e^{-at}}
n
!
(
s
+
a
)
n
+
1
{\displaystyle n! \over {(s+a)^{n+1}}}
cos
w
0
t
{\displaystyle \cos w_{0}t}
s
s
2
+
w
0
2
{\displaystyle s \over {s^{2}+w_{0}^{2}}}
sin
w
0
t
{\displaystyle \sin w_{0}t}
w
0
s
2
+
w
0
2
{\displaystyle w_{0} \over {s^{2}+w_{0}^{2}}}
e
−
a
t
cos
w
0
t
{\displaystyle e^{-at}\cos w_{0}t}
s
+
a
(
s
+
a
)
2
+
w
0
2
{\displaystyle s+a \over {(s+a)^{2}+w_{0}^{2}}}
e
−
a
t
sin
w
0
t
{\displaystyle e^{-at}\sin w_{0}t}
w
0
(
s
+
a
)
2
+
w
0
2
{\displaystyle w_{0} \over {(s+a)^{2}+w_{0}^{2}}}
t
cos
w
0
t
{\displaystyle t\cos w_{0}t}
s
2
−
w
0
2
(
s
2
+
w
0
2
)
2
{\displaystyle s^{2}-w_{0}^{2} \over {(s^{2}+w_{0}^{2})^{2}}}
t
sin
w
0
t
{\displaystyle t\sin w_{0}t}
2
w
0
s
(
s
2
+
w
0
2
)
2
{\displaystyle 2w_{0}s \over {(s^{2}+w_{0}^{2})^{2}}}
Nr.
f
(
x
)
=
1
2
π
j
∫
+
j
∞
−
j
∞
F
(
w
)
e
j
w
t
d
w
{\displaystyle f(x)={{1 \over {2\pi j}}\int _{+j\infty }^{-j\infty }F(w)e^{jwt}dw}}
F
(
w
)
=
∫
−
∞
∞
f
(
t
)
e
−
j
w
t
d
t
{\displaystyle F(w)=\int _{-\infty }^{\infty }f(t)e^{-jwt}dt}
1
δ
(
t
)
{\displaystyle \delta (t)}
1
{\displaystyle 1}
2
h
(
t
)
{\displaystyle h(t)}
1
j
w
+
π
δ
(
w
)
{\displaystyle {1 \over {jw}}+\pi \delta (w)}
3
sin
(
t
)
{\displaystyle \sin(t)}
2
2
w
{\displaystyle 2 \over 2w}
4
1
{\displaystyle 1}
2
π
δ
(
w
)
{\displaystyle 2\pi \delta (w)}
5
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})\}}
6
sin
w
0
t
{\displaystyle \sin w_{0}t}
π
j
{
δ
(
w
−
w
0
)
−
δ
(
w
+
w
0
)
}
{\displaystyle {\pi \over j}\{\delta (w-w_{0})-\delta (w+w_{0})\}}
7
h
(
t
)
cos
w
0
t
{\displaystyle h(t)\cos w_{0}t}
π
2
{
δ
(
w
−
w
0
)
+
δ
(
w
+
w
0
)
}
+
j
w
w
0
2
+
(
j
w
)
2
{\displaystyle {\pi \over 2}\{\delta (w-w_{0})+\delta (w+w_{0})\}+{jw \over {w_{0}^{2}+(jw)^{2}}}}
8
h
(
t
)
cos
w
0
t
{\displaystyle h(t)\cos w_{0}t}
π
2
j
{
δ
(
w
−
w
0
)
+
δ
(
w
+
w
0
)
}
+
w
0
w
0
2
+
(
j
w
)
2
{\displaystyle {\pi \over 2j}\{\delta (w-w_{0})+\delta (w+w_{0})\}+{w_{0} \over {w_{0}^{2}+(jw)^{2}}}}
9
p
α
(
t
)
,
sin
α
t
π
t
{\displaystyle {p_{\alpha }(t)},{\sin \alpha t \over {\pi t}}}
2
sin
w
α
w
,
p
α
(
t
)
{\displaystyle {2\sin w\alpha \over w},{p_{\alpha }(t)}}
10
e
−
a
|
t
|
{\displaystyle e^{-a|t|}}
2
a
a
2
+
w
2
{\displaystyle 2a \over {a^{2}+w^{2}}}
11
1
a
2
+
t
2
{\displaystyle 1 \over {a^{2}+t^{2}}}
π
a
e
−
a
|
w
|
{\displaystyle {\pi \over a}e^{-a|w|}}