Function
Fourier transform unitary, ordinary frequency
Fourier transform unitary, angular frequency
Fourier transform non-unitary, angular frequency
f
(
x
)
{\displaystyle \displaystyle f(x)\,}
f
^
(
ξ
)
=
{\displaystyle \displaystyle {\hat {f}}(\xi )=}
∫
−
∞
∞
f
(
x
)
e
−
2
π
i
x
ξ
d
x
{\displaystyle \displaystyle \int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx}
f
^
(
ν
)
=
{\displaystyle \displaystyle {\hat {f}}(\nu )=}
1
2
π
∫
−
∞
∞
f
(
x
)
e
−
i
ν
x
d
x
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx}
f
^
(
ω
)
=
{\displaystyle \displaystyle {\hat {f}}(\omega )=}
∫
−
∞
∞
f
(
x
)
e
−
i
ω
x
d
x
{\displaystyle \displaystyle \int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx}
Functional relationships
101
a
⋅
f
(
x
)
+
b
⋅
g
(
x
)
{\displaystyle \displaystyle a\cdot f(x)+b\cdot g(x)\,}
a
⋅
f
^
(
ξ
)
+
b
⋅
g
^
(
ξ
)
{\displaystyle \displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,}
a
⋅
f
^
(
ν
)
+
b
⋅
g
^
(
ν
)
{\displaystyle \displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,}
a
⋅
f
^
(
ω
)
+
b
⋅
g
^
(
ω
)
{\displaystyle \displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,}
102
f
(
x
−
a
)
{\displaystyle \displaystyle f(x-a)\,}
e
−
2
π
i
a
ξ
f
^
(
ξ
)
{\displaystyle \displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,}
e
−
i
a
ν
f
^
(
ν
)
{\displaystyle \displaystyle e^{-ia\nu }{\hat {f}}(\nu )\,}
e
−
i
a
ω
f
^
(
ω
)
{\displaystyle \displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,}
103
e
2
π
i
a
x
f
(
x
)
{\displaystyle \displaystyle e^{2\pi iax}f(x)\,}
f
^
(
ξ
−
a
)
{\displaystyle \displaystyle {\hat {f}}\left(\xi -a\right)\,}
f
^
(
ν
−
2
π
a
)
{\displaystyle \displaystyle {\hat {f}}(\nu -2\pi a)\,}
f
^
(
ω
−
2
π
a
)
{\displaystyle \displaystyle {\hat {f}}(\omega -2\pi a)\,}
104
f
(
a
x
)
{\displaystyle \displaystyle f(ax)\,}
1
|
a
|
f
^
(
ξ
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,}
1
|
a
|
f
^
(
ν
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,}
1
|
a
|
f
^
(
ω
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}
105
f
^
(
x
)
{\displaystyle \displaystyle {\hat {f}}(x)\,}
f
(
−
ξ
)
{\displaystyle \displaystyle f(-\xi )\,}
f
(
−
ν
)
{\displaystyle \displaystyle f(-\nu )\,}
2
π
f
(
−
ω
)
{\displaystyle \displaystyle 2\pi f(-\omega )\,}
106
d
n
f
(
x
)
d
x
n
{\displaystyle \displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}
(
2
π
i
ξ
)
n
f
^
(
ξ
)
{\displaystyle \displaystyle (2\pi i\xi )^{n}{\hat {f}}(\xi )\,}
(
i
ν
)
n
f
^
(
ν
)
{\displaystyle \displaystyle (i\nu )^{n}{\hat {f}}(\nu )\,}
107
x
n
f
(
x
)
{\displaystyle \displaystyle x^{n}f(x)\,}
(
i
2
π
)
n
d
n
f
^
(
ξ
)
d
ξ
n
{\displaystyle \displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,}
i
n
d
n
f
^
(
ν
)
d
ν
n
{\displaystyle \displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\nu )}{d\nu ^{n}}}}
i
n
d
n
f
^
(
ω
)
d
ω
n
{\displaystyle \displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}}
108
(
f
∗
g
)
(
x
)
{\displaystyle \displaystyle (f*g)(x)\,}
f
^
(
ξ
)
g
^
(
ξ
)
{\displaystyle \displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,}
2
π
f
^
(
ν
)
g
^
(
ν
)
{\displaystyle \displaystyle {\sqrt {2\pi }}{\hat {f}}(\nu ){\hat {g}}(\nu )\,}
f
^
(
ω
)
g
^
(
ω
)
{\displaystyle \displaystyle {\hat {f}}(\omega ){\hat {g}}(\omega )\,}
109
f
(
x
)
g
(
x
)
{\displaystyle \displaystyle f(x)g(x)\,}
(
f
^
∗
g
^
)
(
ξ
)
{\displaystyle \displaystyle ({\hat {f}}*{\hat {g}})(\xi )\,}
(
f
^
∗
g
^
)
(
ν
)
2
π
{\displaystyle \displaystyle ({\hat {f}}*{\hat {g}})(\nu ) \over {\sqrt {2\pi }}\,}
1
2
π
(
f
^
∗
g
^
)
(
ω
)
{\displaystyle \displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\omega )\,}
110
For
f
(
x
)
{\displaystyle \displaystyle f(x)\,}
a purely real
f
^
(
−
ξ
)
=
f
^
(
ξ
)
¯
{\displaystyle \displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,}
f
^
(
−
ν
)
=
f
^
(
ν
)
¯
{\displaystyle \displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,}
f
^
(
−
ω
)
=
f
^
(
ω
)
¯
{\displaystyle \displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}
111
For
f
(
x
)
{\displaystyle \displaystyle f(x)\,}
a purely real even function
f
^
(
ν
)
{\displaystyle \displaystyle {\hat {f}}(\nu )}
,
f
^
(
ξ
)
{\displaystyle \displaystyle {\hat {f}}(\xi )}
and
f
^
(
ω
)
{\displaystyle \displaystyle {\hat {f}}(\omega )\,}
are purely real even functions.
112
For
f
(
x
)
{\displaystyle \displaystyle f(x)\,}
a purely real odd function
f
^
(
ν
)
{\displaystyle \displaystyle {\hat {f}}(\nu )}
,
f
^
(
ξ
)
{\displaystyle \displaystyle {\hat {f}}(\xi )}
and
f
^
(
ω
)
{\displaystyle \displaystyle {\hat {f}}(\omega )}
are purely imaginary odd functions.
Square-integrable functions
201
rect
(
a
x
)
{\displaystyle \displaystyle \operatorname {rect} (ax)\,}
1
|
a
|
⋅
sinc
(
ξ
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\xi }{a}}\right)}
1
2
π
a
2
⋅
sinc
(
ν
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} \left({\frac {\nu }{2\pi a}}\right)}
1
|
a
|
⋅
sinc
(
ω
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} \left({\frac {\omega }{2\pi a}}\right)}
202
sinc
(
a
x
)
{\displaystyle \displaystyle \operatorname {sinc} (ax)\,}
1
|
a
|
⋅
rect
(
ξ
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\xi }{a}}\right)\,}
1
2
π
a
2
⋅
rect
(
ν
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {rect} \left({\frac {\nu }{2\pi a}}\right)}
1
|
a
|
⋅
rect
(
ω
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {rect} \left({\frac {\omega }{2\pi a}}\right)}
203
sinc
2
(
a
x
)
{\displaystyle \displaystyle \operatorname {sinc} ^{2}(ax)}
1
|
a
|
⋅
tri
(
ξ
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\xi }{a}}\right)}
1
2
π
a
2
⋅
tri
(
ν
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {tri} \left({\frac {\nu }{2\pi a}}\right)}
1
|
a
|
⋅
tri
(
ω
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {tri} \left({\frac {\omega }{2\pi a}}\right)}
204
tri
(
a
x
)
{\displaystyle \displaystyle \operatorname {tri} (ax)}
1
|
a
|
⋅
sinc
2
(
ξ
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\xi }{a}}\right)\,}
1
2
π
a
2
⋅
sinc
2
(
ν
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \operatorname {sinc} ^{2}\left({\frac {\nu }{2\pi a}}\right)}
1
|
a
|
⋅
sinc
2
(
ω
2
π
a
)
{\displaystyle \displaystyle {\frac {1}{|a|}}\cdot \operatorname {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}
205
e
−
a
x
u
(
x
)
{\displaystyle \displaystyle e^{-ax}u(x)\,}
1
a
+
2
π
i
ξ
{\displaystyle \displaystyle {\frac {1}{a+2\pi i\xi }}}
1
2
π
(
a
+
i
ν
)
{\displaystyle \displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\nu )}}}
1
a
+
i
ω
{\displaystyle \displaystyle {\frac {1}{a+i\omega }}}
206
e
−
α
x
2
{\displaystyle \displaystyle e^{-\alpha x^{2}}\,}
π
α
⋅
e
−
(
π
ξ
)
2
α
{\displaystyle \displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}
1
2
α
⋅
e
−
ν
2
4
α
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\nu ^{2}}{4\alpha }}}}
π
α
⋅
e
−
ω
2
4
α
{\displaystyle \displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}}
207
e
−
a
|
x
|
{\displaystyle \displaystyle \operatorname {e} ^{-a|x|}\,}
2
a
a
2
+
4
π
2
ξ
2
{\displaystyle \displaystyle {\frac {2a}{a^{2}+4\pi ^{2}\xi ^{2}}}}
2
π
⋅
a
a
2
+
ν
2
{\displaystyle \displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\nu ^{2}}}}
2
a
a
2
+
ω
2
{\displaystyle \displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}}
208
J
n
(
x
)
x
{\displaystyle \displaystyle {\frac {J_{n}(x)}{x}}\,}
2
i
n
(
−
i
)
n
⋅
U
n
−
1
(
2
π
ξ
)
{\displaystyle \displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi \xi )\,}
⋅
1
−
4
π
2
ξ
2
rect
(
π
ξ
)
{\displaystyle \displaystyle \cdot \ {\sqrt {1-4\pi ^{2}\xi ^{2}}}\operatorname {rect} (\pi \xi )}
2
π
i
n
(
−
i
)
n
⋅
U
n
−
1
(
ν
)
{\displaystyle \displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\nu )\,}
⋅
1
−
ν
2
rect
(
ν
2
)
{\displaystyle \displaystyle \cdot \ {\sqrt {1-\nu ^{2}}}\operatorname {rect} \left({\frac {\nu }{2}}\right)}
2
i
n
(
−
i
)
n
⋅
U
n
−
1
(
ω
)
{\displaystyle \displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,}
⋅
1
−
ω
2
rect
(
ω
2
)
{\displaystyle \displaystyle \cdot \ {\sqrt {1-\omega ^{2}}}\operatorname {rect} \left({\frac {\omega }{2}}\right)}
209
sech
(
a
x
)
{\displaystyle \displaystyle \operatorname {sech} (ax)\,}
π
a
sech
(
π
2
a
ξ
)
{\displaystyle \displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi ^{2}}{a}}\xi \right)}
1
a
π
2
sech
(
π
2
a
ν
)
{\displaystyle \displaystyle {\frac {1}{a}}{\sqrt {\frac {\pi }{2}}}\operatorname {sech} \left({\frac {\pi }{2a}}\nu \right)}
π
a
sech
(
π
2
a
ω
)
{\displaystyle \displaystyle {\frac {\pi }{a}}\operatorname {sech} \left({\frac {\pi }{2a}}\omega \right)}
Distributions
301
1
{\displaystyle \displaystyle 1}
δ
(
ξ
)
{\displaystyle \displaystyle \delta (\xi )}
2
π
⋅
δ
(
ν
)
{\displaystyle \displaystyle {\sqrt {2\pi }}\cdot \delta (\nu )}
2
π
δ
(
ω
)
{\displaystyle \displaystyle 2\pi \delta (\omega )}
302
δ
(
x
)
{\displaystyle \displaystyle \delta (x)\,}
1
{\displaystyle \displaystyle 1}
1
2
π
{\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi }}}\,}
1
{\displaystyle \displaystyle 1}
303
e
i
a
x
{\displaystyle \displaystyle e^{iax}}
δ
(
ξ
−
a
2
π
)
{\displaystyle \displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)}
2
π
⋅
δ
(
ν
−
a
)
{\displaystyle \displaystyle {\sqrt {2\pi }}\cdot \delta (\nu -a)}
2
π
δ
(
ω
−
a
)
{\displaystyle \displaystyle 2\pi \delta (\omega -a)}
304
cos
(
a
x
)
{\displaystyle \displaystyle \cos(ax)}
δ
(
ξ
−
a
2
π
)
+
δ
(
ξ
+
a
2
π
)
2
{\displaystyle \displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)+\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2}}}
2
π
⋅
δ
(
ν
−
a
)
+
δ
(
ν
+
a
)
2
{\displaystyle \displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\nu -a)+\delta (\nu +a)}{2}}\,}
π
(
δ
(
ω
−
a
)
+
δ
(
ω
+
a
)
)
{\displaystyle \displaystyle \pi \left(\delta (\omega -a)+\delta (\omega +a)\right)}
305
sin
(
a
x
)
{\displaystyle \displaystyle \sin(ax)}
δ
(
ξ
−
a
2
π
)
−
δ
(
ξ
+
a
2
π
)
2
i
{\displaystyle \displaystyle {\frac {\displaystyle \delta \left(\xi -{\frac {a}{2\pi }}\right)-\delta \left(\xi +{\frac {a}{2\pi }}\right)}{2i}}}
2
π
⋅
δ
(
ν
−
a
)
−
δ
(
ν
+
a
)
2
i
{\displaystyle \displaystyle {\sqrt {2\pi }}\cdot {\frac {\delta (\nu -a)-\delta (\nu +a)}{2i}}}
−
i
π
(
δ
(
ω
−
a
)
−
δ
(
ω
+
a
)
)
{\displaystyle \displaystyle -i\pi \left(\delta (\omega -a)-\delta (\omega +a)\right)}
306
cos
(
a
x
2
)
{\displaystyle \displaystyle \cos(ax^{2})}
π
a
cos
(
π
2
ξ
2
a
−
π
4
)
{\displaystyle \displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}
1
2
a
cos
(
ν
2
4
a
−
π
4
)
{\displaystyle \displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}
π
a
cos
(
ω
2
4
a
−
π
4
)
{\displaystyle \displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}
307
sin
(
a
x
2
)
{\displaystyle \displaystyle \sin(ax^{2})\,}
−
π
a
sin
(
π
2
ξ
2
a
−
π
4
)
{\displaystyle \displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}\xi ^{2}}{a}}-{\frac {\pi }{4}}\right)}
−
1
2
a
sin
(
ν
2
4
a
−
π
4
)
{\displaystyle \displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\nu ^{2}}{4a}}-{\frac {\pi }{4}}\right)}
−
π
a
sin
(
ω
2
4
a
−
π
4
)
{\displaystyle \displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}
308
x
n
{\displaystyle \displaystyle x^{n}\,}
(
i
2
π
)
n
δ
(
n
)
(
ξ
)
{\displaystyle \displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(\xi )\,}
i
n
2
π
δ
(
n
)
(
ν
)
{\displaystyle \displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\nu )\,}
2
π
i
n
δ
(
n
)
(
ω
)
{\displaystyle \displaystyle 2\pi i^{n}\delta ^{(n)}(\omega )\,}
309
1
x
{\displaystyle \displaystyle {\frac {1}{x}}}
−
i
π
sgn
(
ξ
)
{\displaystyle \displaystyle -i\pi \operatorname {sgn}(\xi )}
−
i
π
2
sgn
(
ν
)
{\displaystyle \displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\nu )}
−
i
π
sgn
(
ω
)
{\displaystyle \displaystyle -i\pi \operatorname {sgn}(\omega )}
310
1
x
n
:=
(
−
1
)
n
−
1
(
n
−
1
)
!
d
n
d
x
n
log
|
x
|
{\displaystyle \displaystyle {\frac {1}{x^{n}}}:={\frac {(-1)^{n-1}}{(n-1)!}}{\frac {d^{n}}{dx^{n}}}\log |x|}
−
i
π
(
−
2
π
i
ξ
)
n
−
1
(
n
−
1
)
!
sgn
(
ξ
)
{\displaystyle \displaystyle -i\pi {\frac {(-2\pi i\xi )^{n-1}}{(n-1)!}}\operatorname {sgn}(\xi )}
−
i
π
2
⋅
(
−
i
ν
)
n
−
1
(
n
−
1
)
!
sgn
(
ν
)
{\displaystyle \displaystyle -i{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\nu )^{n-1}}{(n-1)!}}\operatorname {sgn}(\nu )}
−
i
π
(
−
i
ω
)
n
−
1
(
n
−
1
)
!
sgn
(
ω
)
{\displaystyle \displaystyle -i\pi {\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )}
311
|
x
|
α
{\displaystyle \displaystyle |x|^{\alpha }\,}
−
2
sin
(
π
α
/
2
)
Γ
(
α
+
1
)
|
2
π
ξ
|
α
+
1
{\displaystyle \displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|2\pi \xi |^{\alpha +1}}}}
−
2
2
π
sin
(
π
α
/
2
)
Γ
(
α
+
1
)
|
ν
|
α
+
1
{\displaystyle \displaystyle {\frac {-2}{\sqrt {2\pi }}}{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}}
−
2
sin
(
π
α
/
2
)
Γ
(
α
+
1
)
|
ν
|
α
+
1
{\displaystyle \displaystyle -2{\frac {\sin(\pi \alpha /2)\Gamma (\alpha +1)}{|\nu |^{\alpha +1}}}}
312
sgn
(
x
)
{\displaystyle \displaystyle \operatorname {sgn}(x)}
1
i
π
ξ
{\displaystyle \displaystyle {\frac {1}{i\pi \xi }}}
2
π
⋅
1
i
ν
{\displaystyle \displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\nu }}\,}
2
i
ω
{\displaystyle \displaystyle {\frac {2}{i\omega }}}
313
u
(
x
)
{\displaystyle \displaystyle u(x)}
1
2
(
1
i
π
ξ
+
δ
(
ξ
)
)
{\displaystyle \displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi \xi }}+\delta (\xi )\right)}
π
2
(
1
i
π
ν
+
δ
(
ν
)
)
{\displaystyle \displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \nu }}+\delta (\nu )\right)}
π
(
1
i
π
ω
+
δ
(
ω
)
)
{\displaystyle \displaystyle \pi \left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)}
314
∑
n
=
−
∞
∞
δ
(
x
−
n
T
)
{\displaystyle \displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nT)}
1
T
∑
k
=
−
∞
∞
δ
(
ξ
−
k
T
)
{\displaystyle \displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\xi -{\frac {k}{T}}\right)}
2
π
T
∑
k
=
−
∞
∞
δ
(
ν
−
2
π
k
T
)
{\displaystyle \displaystyle {\frac {\sqrt {2\pi }}{T}}\sum _{k=-\infty }^{\infty }\delta \left(\nu -{\frac {2\pi k}{T}}\right)}
2
π
T
∑
k
=
−
∞
∞
δ
(
ω
−
2
π
k
T
)
{\displaystyle \displaystyle {\frac {2\pi }{T}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -{\frac {2\pi k}{T}}\right)}
315
J
0
(
x
)
{\displaystyle \displaystyle J_{0}(x)}
2
rect
(
π
ξ
)
1
−
4
π
2
ξ
2
{\displaystyle \displaystyle {\frac {2\,\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}
2
π
⋅
rect
(
ν
2
)
1
−
ν
2
{\displaystyle \displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}
2
rect
(
ω
2
)
1
−
ω
2
{\displaystyle \displaystyle {\frac {2\,\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}
316
J
n
(
x
)
{\displaystyle \displaystyle J_{n}(x)}
2
(
−
i
)
n
T
n
(
2
π
ξ
)
rect
(
π
ξ
)
1
−
4
π
2
ξ
2
{\displaystyle \displaystyle {\frac {2(-i)^{n}T_{n}(2\pi \xi )\operatorname {rect} (\pi \xi )}{\sqrt {1-4\pi ^{2}\xi ^{2}}}}}
2
π
(
−
i
)
n
T
n
(
ν
)
rect
(
ν
2
)
1
−
ν
2
{\displaystyle \displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\nu )\operatorname {rect} \left(\displaystyle {\frac {\nu }{2}}\right)}{\sqrt {1-\nu ^{2}}}}}
2
(
−
i
)
n
T
n
(
ω
)
rect
(
ω
2
)
1
−
ω
2
{\displaystyle \displaystyle {\frac {2(-i)^{n}T_{n}(\omega )\operatorname {rect} \left(\displaystyle {\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}