L
[
a
f
(
t
)
]
=
a
F
(
s
)
L[af(t)]=aF(s)
L[af(t)]=aF(s)
线性定理 - 叠加性
L
(
f
1
(
t
)
±
f
2
(
t
)
)
=
F
1
(
s
)
±
F
2
(
s
)
L(f_1(t)\pm f_2(t))=F_1(s)\pm F_2(s)
L(f1(t)±f2(t))=F1(s)±F2(s)
微分定理 - 一阶导
L
[
d
f
(
t
)
d
t
]
=
s
F
(
s
)
−
f
(
0
)
L[\frac{df(t)}{dt}]=sF(s)-f(0)
L[dtdf(t)]=sF(s)−f(0)
微分定理 - 二阶导
L
[
d
2
f
(
t
)
d
t
2
]
=
s
2
F
(
s
)
−
s
f
(
0
)
−
f
′
(
0
)
L[\frac{d^2f(t)}{dt^2}]=s^2F(s)-sf(0)-f'(0)
L[dt2d2f(t)]=s2F(s)−sf(0)−f′(0)
微分定理 - n阶导
L
[
d
n
f
(
t
)
d
t
n
]
=
s
n
F
(
s
)
−
∑
k
=
1
n
s
n
−
k
f
k
−
1
(
0
)
L[\frac{d^n f(t)}{dt^n}]=s^nF(s)-\sum_{k=1}^{n}s^{n-k}f^{k-1}(0)
L[dtndnf(t)]=snF(s)−∑k=1nsn−kfk−1(0)
微分定理
L
[
t
f
(
t
)
]
=
−
d
d
s
F
(
s
)
L[tf(t)]=-\frac{d}{ds}F(s)
L[tf(t)]=−dsdF(s)
积分定理 - 一阶导
L
[
∫
f
(
t
)
d
t
]
=
F
(
s
)
s
+
[
∫
f
(
t
)
d
t
]
t
=
0
s
L[\int f(t)dt]=\frac{F(s)}{s}+\frac{[\int f(t)dt]_{t=0}}{s}
L[∫f(t)dt]=sF(s)+s[∫f(t)dt]t=0
积分定理 - 二阶导
L
[
∬
f
(
t
)
(
d
t
)
2
]
=
F
(
s
)
s
2
+
[
∫
f
(
t
)
d
t
]
t
=
0
s
2
+
[
∬
f
(
t
)
(
d
t
)
2
]
t
=
0
s
L[\iint f(t)(dt)^2]=\frac{F(s)}{s^2}+\frac{[\int f(t)dt]_{t=0}}{s^2}+\frac{[\iint f(t)(dt)^2]_{t=0}}{s}
L[∬f(t)(dt)2]=s2F(s)+s2[∫f(t)dt]t=0+s[∬f(t)(dt)2]t=0
积分定理 - n阶导
L
[
∫
…
∫
⏞
n
f
(
t
)
(
d
t
)
n
]
=
F
(
s
)
s
n
+
∑
k
=
1
n
[
∫
…
∫
⏞
k
f
(
t
)
(
d
t
)
k
]
t
=
0
s
n
−
k
+
1
L[\overbrace{\int \dotso \int}^{n}f(t)(dt)^n]=\frac{F(s)}{s^n}+\sum_{k=1}^n\frac{[\overbrace{\int \dotso \int}^{k}f(t)(dt)^k]_{t=0}}{s^{n-k+1}}
L[∫…∫nf(t)(dt)n]=snF(s)+∑k=1nsn−k+1[∫…∫kf(t)(dt)k]t=0
延迟定理
L
[
f
(
t
−
T
)
1
(
t
−
T
)
]
=
e
−
T
s
F
(
s
)
L[f(t-T)1(t-T)]=e^{-Ts}F(s)
L[f(t−T)1(t−T)]=e−TsF(s)
衰减定理
L
[
f
(
t
)
e
−
a
t
]
=
F
(
s
+
a
)
L[f(t)e^{-at}]=F(s+a)
L[f(t)e−at]=F(s+a)
终值定理
lim
t
→
∞
f
(
t
)
=
lim
s
→
0
s
F
(
s
)
\lim\limits_{t \to \infty}f(t)=\lim\limits_{s \to 0}sF(s)
t→∞limf(t)=s→0limsF(s)
初值定理
lim
t
→
0
f
(
t
)
=
lim
s
→
∞
s
F
(
s
)
\lim\limits_{t \to 0}f(t)=\lim\limits_{s \to \infty}sF(s)
t→0limf(t)=s→∞limsF(s)
卷积定理
L
[
∫
0
t
f
1
(
t
−
τ
)
f
2
(
τ
)
d
τ
]
=
F
1
(
s
)
F
2
(
s
)
L[\int_{0}^{t}f_1(t-\tau)f_2(\tau)d\tau]=F_1(s)F_2(s)
L[∫0tf1(t−τ)f2(τ)dτ]=F1(s)F2(s)
尺度定理
L
[
f
(
a
t
)
]
=
1
∣
a
∣
f
(
s
a
)
L[f(at)]=\frac{1}{\vert a\vert}f(\frac{s}{a})
L[f(at)]=∣a∣1f(as)
常用函数的变换
时间函数
变换后
时间函数
变换后
δ
(
t
)
\delta(t)
δ(t)
1
1
−
e
−
a
t
1-e^{-at}
1−e−at
a
s
(
s
+
a
)
\frac{a}{s(s+a)}
s(s+a)a
δ
T
(
t
)
=
∑
n
=
0
∞
δ
(
t
−
n
T
)
\delta_T(t)=\sum_{n=0}^\infty\delta(t-nT)
δT(t)=∑n=0∞δ(t−nT)
1
1
−
e
−
T
s
\frac{1}{1-e^{-Ts}}
1−e−Ts1
e
−
a
t
−
e
−
b
t
e^{-at}-e^{-bt}
e−at−e−bt
b
−
a
(
s
+
a
)
(
s
+
b
)
\frac{b-a}{(s+a)(s+b)}
(s+a)(s+b)b−a
1
(
t
)
1(t)
1(t)
1
s
\frac{1}{s}
s1
sin
ω
t
\sin \omega t
sinωt
ω
s
2
+
ω
2
\frac{\omega}{s^2+\omega^2}
s2+ω2ω
t
t
t
1
s
2
\frac{1}{s^2}
s21
cos
ω
t
\cos \omega t
cosωt
s
s
2
+
ω
2
\frac{s}{s^2+\omega^2}
s2+ω2s
t
2
2
\frac{t^2}{2}
2t2
1
s
3
\frac{1}{s^3}
s31
e
−
a
t
sin
ω
t
e^{-at}\sin \omega t
e−atsinωt
ω
(
s
+
a
)
2
+
ω
2
\frac{\omega}{(s+a)^2+\omega^2}
(s+a)2+ω2ω
t
n
n
\frac{t^n}{n}
ntn
1
s
n
+
1
\frac{1}{s^{n+1}}
sn+11
e
−
a
t
cos
ω
t
e^{-at}\cos \omega t
e−atcosωt
s
+
a
(
s
+
a
)
2
+
ω
2
\frac{s+a}{(s+a)^2+\omega^2}
(s+a)2+ω2s+a
e
−
a
t
e^{-at}
e−at
1
s
+
a
\frac{1}{s+a}
s+a1
a
t
/
T
a^{t/T}
at/T
1
s
−
(
1
/
t
)
ln
a
\frac{1}{s-(1/t)\ln a}
s−(1/t)lna1