如果
f
(
t
)
f(t)
f(t)满足：在
(
−
∞
,
+
∞
)
( - \infty , + \infty )
(−∞,+∞)上连续或只有有限个可去间断点，且当
∣
t
∣
→
+
∞
|t| \to + \infty
∣t∣→+∞时，
f
(
t
)
f(t)
f(t)
→
\to
→
0
0
0,则
F
[
f
′
(
t
)
]
=
j
ω
F
[
f
(
t
)
]
\mathscr F[f'(t)]=j\omega \mathscr F[f(t)]
F[f′(t)]=jωF[f(t)]
该式表明一个函数导数的Fourier变换等于这个函数的Fourier变换乘以
j
ω
j\omega
jω.
证明：由Fourier变换的定义, 并利用分部积分可得
F
[
f
′
(
t
)
]
=
∫
−
∞
+
∞
f
′
(
t
)
e
−
j
ω
t
d
t
=
f
(
t
)
e
−
j
ω
t
∣
−
∞
+
∞
+
j
ω
∫
−
∞
+
∞
f
(
t
)
e
−
j
ω
t
d
t
=
j
ω
F
[
f
(
t
)
]
\mathscr F[f'(t)] = \int_{ - \infty }^{ + \infty } {f'(t){e^{ - j\omega t}}dt} \\ \\ \\= \left. {f(t){e^{ - j\omega t}}} \right|_{ - \infty }^{ + \infty } + j\omega \int_{ - \infty }^{ + \infty } {f(t){e^{ - j\omega t}}dt} \\ \\ \\= j\omega \mathscr F[f(t)]
F[f′(t)]=∫−∞+∞f′(t)e−jωtdt=f(t)e−jωt
−∞+∞+jω∫−∞+∞f(t)e−jωtdt=jωF[f(t)]
注：分部积分公式： ∫ u d v = u v − ∫ v d u \int {udv = uv - \int {vdu} } ∫udv=uv−∫vdu
推论：若 f ( k ) ( t ) f^{(k)}(t) f(k)(t)在 ( − ∞ , + ∞ ) ( - \infty , + \infty ) (−∞,+∞)上连续或只有有限个可去间断点，且 lim ∣ t ∣ → ∞ f ( k ) ( t ) = 0 \lim \limits_{|t| \to \infty } \,{f^{(k)}}(t) = 0 ∣t∣→∞limf(k)(t)=0, k = 0 , 1 , 2 , . . . , n − 1 k=0,1,2,...,n-1 k=0,1,2,...,n−1,则
F [ f ( n ) ( t ) ] = ( j ω ) n F [ f ( t ) ] \mathscr F[{f^{\left( n \right)}}(t)] = {\left( {{\rm{j}}\omega } \right)^n}\mathscr F[f(t)] F[f(n)(t)]=(jω)nF[f(t)]
象函数的导数公式：
设
F
[
f
(
t
)
]
=
F
(
ω
)
\mathscr F[f(t)] = F(\omega )
F[f(t)]=F(ω),则
d
d
ω
F
(
ω
)
=
F
[
−
j
t
f
(
t
)
]
\frac{d}{{d\omega }}F(\omega ) =\mathscr F[ - jtf(t)]
dωdF(ω)=F[−jtf(t)]
该式表明一个函数的象函数的导数等于该函数乘以
−
j
t
-jt
−jt后的Fourier变换。
一般地，
d
n
d
ω
n
F
(
ω
)
=
(
−
j
)
n
F
[
t
n
f
(
t
)
]
\frac{{{d^n}}}{{d{\omega ^n}}}F(\omega ) = {( - j)^n}\mathscr F[{t^n}f(t)]
dωndnF(ω)=(−j)nF[tnf(t)]