Editing Fourier transform (section)

== Tables of important Fourier transforms ==
The following tables record some closed-form Fourier transforms. For functions {{math|''f''(''x'')}} and {{math|''g''(''x'')}} denote their Fourier transforms by {{math|''f̂''}} and {{math|''ĝ''}}. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.

=== Functional relationships, one-dimensional ===
The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}.
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !!Remarks
|-
|
|<math> f(x)\,</math>
|<math>\begin{align} &\widehat{f}(\xi) \triangleq \widehat {f_1}(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_2}(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\widehat{f}(\omega) \triangleq \widehat {f_3}(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
| 101
|<math> a\, f(x) + b\, g(x)\,</math>
|<math> a\, \widehat{f}(\xi) + b\, \widehat{g}(\xi)\,</math>
|<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>
|<math> a\, \widehat{f}(\omega) + b\, \widehat{g}(\omega)\,</math>
|Linearity
|-
| 102
|<math> f(x - a)\,</math>
|<math> e^{-i 2\pi \xi a} \widehat{f}(\xi)\,</math>
|<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>
|<math> e^{- i a \omega} \widehat{f}(\omega)\,</math>
|Shift in time domain
|-
| 103
|<math> f(x)e^{iax}\,</math>
|<math> \widehat{f} \left(\xi - \frac{a}{2\pi}\right)\,</math>
|<math> \widehat{f}(\omega - a)\,</math>
|<math> \widehat{f}(\omega - a)\,</math>
|Shift in frequency domain, dual of 102
|-
| 104
|<math> f(a x)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\xi}{a} \right)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>
|<math> \frac{1}{|a|} \widehat{f}\left( \frac{\omega}{a} \right)\,</math>
|Scaling in the time domain. If {{math|{{abs|''a''}}}} is large, then {{math|''f''(''ax'')}} is concentrated around 0 and{{br}}<math> \frac{1}{|a|}\hat{f} \left( \frac{\omega}{a} \right)\,</math>{{br}}spreads out and flattens.
|-
| 105
|<math> \widehat {f_n}(x)\,</math>
|<math> \widehat {f_1}(x) \ \stackrel{\mathcal{F}_1}{\longleftrightarrow}\  f(-\xi)\,</math>
|<math> \widehat {f_2}(x) \ \stackrel{\mathcal{F}_2}{\longleftrightarrow}\  f(-\omega)\,</math>
|<math> \widehat {f_3}(x) \ \stackrel{\mathcal{F}_3}{\longleftrightarrow}\ 2\pi f(-\omega)\,</math>
|The same transform is applied twice, but ''x'' replaces the frequency variable (''ξ'' or ''ω'') after the first transform.
|-
| 106
|<math> \frac{d^n f(x)}{dx^n}\,</math>
|<math> (i 2\pi \xi)^n \widehat{f}(\xi)\,</math>
|<math> (i\omega)^n \widehat{f}(\omega)\,</math>
|<math> (i\omega)^n \widehat{f}(\omega)\,</math>
|n{{superscript|th}}-order derivative.
As {{math|''f''}} is a [[Schwartz space|Schwartz function]]
|-
|106.5
|<math>\int_{-\infty}^{x} f(\tau) d \tau</math>
|<math>\frac{\widehat{f}(\xi)}{i 2 \pi \xi} + C \, \delta(\xi)</math>
|<math>\frac{\widehat{f} (\omega)}{i\omega} + \sqrt{2 \pi} C \delta(\omega)</math>
|<math>\frac{\widehat{f} (\omega)}{i\omega} + 2 \pi C \delta(\omega)</math>
|Integration.<ref>{{Cite web |date=2015 |orig-date=2010 |title=The Integration Property of the Fourier Transform |url=https://www.thefouriertransform.com/transform/integration.php |url-status=live |archive-url=https://web.archive.org/web/20220126171340/https://www.thefouriertransform.com/transform/integration.php |archive-date=2022-01-26 |access-date=2023-08-20 |website=The Fourier Transform .com}}</ref> Note: <math>\delta</math> is the [[Dirac delta function]] and <math>C</math> is the average ([[DC component|DC]]) value of <math>f(x)</math> such that <math>\int_{-\infty}^\infty (f(x) - C) \, dx = 0</math>
|-
| 107
|<math> x^n f(x)\,</math>
|<math> \left (\frac{i}{2\pi}\right)^n \frac{d^n \widehat{f}(\xi)}{d\xi^n}\,</math>
|<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>
|<math> i^n \frac{d^n \widehat{f}(\omega)}{d\omega^n}</math>
|This is the dual of 106
|-
| 108
|<math> (f * g)(x)\,</math>
|<math> \widehat{f}(\xi) \widehat{g}(\xi)\,</math>
|<math> \sqrt{2\pi}\ \widehat{f}(\omega) \widehat{g}(\omega)\,</math>
|<math> \widehat{f}(\omega) \widehat{g}(\omega)\,</math>
|The notation {{math|''f'' ∗ ''g''}} denotes the [[convolution]] of {{mvar|f}} and {{mvar|g}} — this rule is the [[convolution theorem]]
|-
| 109
|<math> f(x) g(x)\,</math>
|<math> \left(\widehat{f} * \widehat{g}\right)(\xi)\,</math>
|<math> \frac{1}\sqrt{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>
|<math> \frac{1}{2\pi}\left(\widehat{f} * \widehat{g}\right)(\omega)\,</math>
|This is the dual of 108
|-
| 110
|For {{math|''f''(''x'')}} purely real
|<math> \widehat{f}(-\xi) = \overline{\widehat{f}(\xi)}\,</math>
|<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>
|<math> \widehat{f}(-\omega) = \overline{\widehat{f}(\omega)}\,</math>
|Hermitian symmetry. {{math|{{overline|''z''}}}} indicates the [[complex conjugate]].
|-
<!-- A Symmetry section has been added instead of this.
| 111
|For {{math|''f''(''x'')}} purely real and [[even function|even]]
| colspan=3 align=center |<math>\widehat f </math> is a purely real and [[even function]].
|
|-
| 112
|For {{math|''f''(''x'')}} purely real and [[odd function|odd]]
| colspan=3 align=center |<math>\widehat f </math> is a purely [[imaginary number|imaginary]] and [[odd function]].
|
|-->
| 113
|For {{math|''f''(''x'')}} purely imaginary
|<math> \widehat{f}(-\xi) = -\overline{\widehat{f}(\xi)}\,</math>
|<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>
|<math> \widehat{f}(-\omega) = -\overline{\widehat{f}(\omega)}\,</math>
|{{math|{{overline|''z''}}}} indicates the [[complex conjugate]].
|-
| 114
| <math> \overline{f(x)}</math>|| <math> \overline{\widehat{f}(-\xi)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || <math> \overline{\widehat{f}(-\omega)}</math> || [[Complex conjugate|Complex conjugation]], generalization of 110 and 113
|-
|115
|<math> f(x) \cos (a x)</math>
|<math> \frac{ \widehat{f}\left(\xi - \frac{a}{2\pi}\right)+\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2}</math>
|<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}\,</math>
|<math> \frac{\widehat{f}(\omega-a)+\widehat{f}(\omega+a)}{2}</math>
|This follows from rules 101 and 103 using [[Euler's formula]]:{{br}}<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
|-
|116
|<math> f(x)\sin( ax)</math>
|<math> \frac{\widehat{f}\left(\xi-\frac{a}{2\pi}\right)-\widehat{f}\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>
|<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>
|<math> \frac{\widehat{f}(\omega-a)-\widehat{f}(\omega+a)}{2i}</math>
|This follows from 101 and 103 using [[Euler's formula]]:{{br}}<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>
|}

=== Square-integrable functions, one-dimensional ===
The Fourier transforms in this table may be found in {{harvtxt|Campbell|Foster|1948}}, {{harvtxt|Erdélyi|1954}}, or {{harvtxt|Kammler|2000|loc=appendix}}.
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|
|<math> f(x)\,</math>
|<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
|{{anchor|rect}} 201
|<math> \operatorname{rect}(a x) \,</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}\left(\frac{\xi}{a}\right)</math>
|<math> \frac{1}{\sqrt{2 \pi a^2}}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|The [[rectangular function|rectangular pulse]] and the ''normalized'' [[sinc function]], here defined as {{math|1=sinc(''x'') = {{sfrac|sin(π''x'')|π''x''}}}}
|-
| 202
|<math> \operatorname{sinc}(a x)\,</math>
|<math> \frac{1}{|a|}\, \operatorname{rect}\left(\frac{\xi}{a} \right)\,</math>
|<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|<math> \frac{1}{|a|}\, \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|Dual of rule 201. The [[rectangular function]] is an ideal [[low-pass filter]], and the [[sinc function]] is the [[Anticausal system|non-causal]] impulse response of such a filter. The [[sinc function]] is defined here as {{math|1=sinc(''x'') = {{sfrac|sin(π''x'')|π''x''}}}}
|-
| 203
|<math> \operatorname{sinc}^2 (a x)</math>
|<math> \frac{1}{|a|}\, \operatorname{tri} \left( \frac{\xi}{a} \right) </math>
|<math> \frac{1}{\sqrt{2\pi a^2}}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
|<math> \frac{1}{|a|}\, \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
| The function {{math|tri(''x'')}} is the [[triangular function]]
|-
| 204
|<math> \operatorname{tri} (a x)</math>
|<math> \frac{1}{|a|}\, \operatorname{sinc}^2 \left( \frac{\xi}{a} \right) \,</math>
|<math> \frac{1}{\sqrt{2\pi a^2}} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
|<math> \frac{1}{|a|} \, \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
| Dual of rule 203.
|-
| 205
|<math> e^{- a x} u(x) \,</math>
|<math> \frac{1}{a + i 2\pi \xi}</math>
|<math> \frac{1}{\sqrt{2 \pi} (a + i \omega)}</math>
|<math> \frac{1}{a + i \omega}</math>
|The function {{math|''u''(''x'')}} is the [[Heaviside step function|Heaviside unit step function]] and {{math|''a'' > 0}}.
|-
| 206
|<math> e^{-\alpha x^2}\,</math>
|<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{(\pi \xi)^2}{\alpha}}</math>
|<math> \frac{1}{\sqrt{2 \alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>
|<math> \sqrt{\frac{\pi}{\alpha}}\, e^{-\frac{\omega^2}{4 \alpha}}</math>
|This shows that, for the unitary Fourier transforms, the [[Gaussian function]] {{math|''e''<sup>−''αx''<sup>2</sup></sup>}} is its own Fourier transform for some choice of {{mvar|α}}. For this to be integrable we must have {{math|Re(''α'') > 0}}.
|-
| 208
|<math> e^{-a|x|} \,</math>
|<math> \frac{2 a}{a^2 + 4 \pi^2 \xi^2} </math>
|<math> \sqrt{\frac{2}{\pi}} \, \frac{a}{a^2 + \omega^2} </math>
|<math> \frac{2a}{a^2 + \omega^{2}} </math>
|For {{math|Re(''a'') > 0}}. That is, the Fourier transform of a [[Laplace distribution|two-sided decaying exponential function]] is a [[Lorentzian function]].
|-
| 209
|<math> \operatorname{sech}(a x) \,</math>
|<math> \frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} \xi \right)</math>
|<math> \frac{1}{a}\sqrt{\frac{\pi}{2}} \operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>
|<math> \frac{\pi}{a}\operatorname{sech}\left( \frac{\pi}{2 a} \omega \right)</math>
|[[Hyperbolic function|Hyperbolic secant]] is its own Fourier transform
|-
| 210
|<math> e^{-\frac{a^2 x^2}2} H_n(a x)\,</math>
|<math> \frac{\sqrt{2\pi}(-i)^n}{a} e^{-\frac{2\pi^2\xi^2}{a^2}} H_n\left(\frac{2\pi\xi}a\right)</math>
|<math> \frac{(-i)^n}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a\right)</math>
|<math> \frac{(-i)^n \sqrt{2\pi}}{a} e^{-\frac{\omega^2}{2 a^2}} H_n\left(\frac \omega a \right)</math>
|{{math|''H<sub>n</sub>''}} is the {{mvar|n}}th-order [[Hermite polynomial]]. If {{math|''a'' {{=}} 1}} then the Gauss–Hermite functions are [[eigenfunction]]s of the Fourier transform operator. For a derivation, see [[Hermite polynomials#Hermite functions as eigenfunctions of the Fourier transform|Hermite polynomial]]. The formula reduces to 206 for {{math|''n'' {{=}} 0}}.
|}

=== Distributions, one-dimensional ===
The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}.
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|
|<math> f(x)\,</math>
|<math>\begin{align} &\hat{f}(\xi) \triangleq \hat f_1(\xi) \\&= \int_{-\infty}^\infty f(x) e^{-i 2\pi \xi x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_2(\omega) \\&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|<math>\begin{align} &\hat{f}(\omega) \triangleq \hat f_3(\omega) \\&= \int_{-\infty}^\infty f(x) e^{-i \omega x}\, dx \end{align}</math>
|Definitions
|-
| 301
|<math> 1</math>
|<math> \delta(\xi)</math>
|<math> \sqrt{2\pi}\, \delta(\omega)</math>
|<math> 2\pi\delta(\omega)</math>
|The distribution {{math|''δ''(''ξ'')}} denotes the [[Dirac delta function]].
|-
| 302
|<math> \delta(x)\,</math>
|<math> 1</math>
|<math> \frac{1}{\sqrt{2\pi}}\,</math>
|<math> 1</math>
|Dual of rule 301.
|-
| 303
|<math> e^{i a x}</math>
|<math> \delta\left(\xi - \frac{a}{2\pi}\right)</math>
|<math> \sqrt{2 \pi}\, \delta(\omega - a)</math>
|<math> 2 \pi\delta(\omega - a)</math>
|This follows from 103 and 301.
|-
| 304
|<math> \cos (a x)</math>
|<math> \frac{ \delta\left(\xi - \frac{a}{2\pi}\right)+\delta\left(\xi+\frac{a}{2\pi}\right)}{2}</math>
|<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)+\delta(\omega+a)}{2}</math>
|<math> \pi\left(\delta(\omega-a)+\delta(\omega+a)\right)</math>
|This follows from rules 101 and 303 using [[Euler's formula]]:{{br}}<math>\cos(a x) = \frac{e^{i a x} + e^{-i a x}}{2}.</math>
|-
| 305
|<math> \sin( ax)</math>
|<math> \frac{\delta\left(\xi-\frac{a}{2\pi}\right)-\delta\left(\xi+\frac{a}{2\pi}\right)}{2i}</math>
|<math> \sqrt{2 \pi}\,\frac{\delta(\omega-a)-\delta(\omega+a)}{2i}</math>
|<math> -i\pi\bigl(\delta(\omega-a)-\delta(\omega+a)\bigr)</math>
|This follows from 101 and 303 using{{br}}<math>\sin(a x) = \frac{e^{i a x} - e^{-i a x}}{2i}.</math>
|-
| 306
|<math> \cos \left( a x^2 \right) </math>
|<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>
|<math> \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right) </math>
|This follows from 101 and 207 using{{br}}<math>\cos(a x^2) = \frac{e^{i a x^2} + e^{-i a x^2}}{2}.</math>
|-
| 307
|<math> \sin \left( a x^2 \right) </math>
|<math> - \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 \xi^2}{a} - \frac{\pi}{4} \right) </math>
|<math> \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> -\sqrt{\frac{\pi}{a}}\sin \left( \frac{\omega^2}{4a} - \frac{\pi}{4} \right)</math>
|This follows from 101 and 207 using{{br}}<math>\sin(a x^2) = \frac{e^{i a x^2} - e^{-i a x^2}}{2i}.</math>
|-
|308
|<math> e^{-\pi i\alpha x^2}\,</math>
|<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\pi \xi^2}{\alpha}}</math>
|<math> \frac{1}{\sqrt{2\pi \alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>
|<math> \frac{1}{\sqrt{\alpha}}\, e^{-i\frac{\pi}{4}} e^{i\frac{\omega^2}{4\pi \alpha}}</math>	
|Here it is assumed <math>\alpha</math> is real. For the case that alpha is complex see table entry 206 above.
|-
| 309
|<math> x^n\,</math>
|<math> \left(\frac{i}{2\pi}\right)^n \delta^{(n)} (\xi)</math>
|<math> i^n \sqrt{2\pi} \delta^{(n)} (\omega)</math>
|<math> 2\pi i^n\delta^{(n)} (\omega)</math>
|Here, {{mvar|n}} is a [[natural number]] and {{math|''δ''{{isup|(''n'')}}(''ξ'')}} is the {{mvar|n}}th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all [[polynomial]]s.
|-
| 310
|<math> \delta^{(n)}(x)</math>
|<math> (i 2\pi \xi)^n</math>
|<math> \frac{(i\omega)^n}{\sqrt{2\pi}} </math>
|<math> (i\omega)^n</math>
|Dual of rule 309. {{math|''δ''{{isup|(''n'')}}(''ξ'')}} is the {{mvar|n}}th distribution derivative of the Dirac delta function. This rule follows from 106 and 302.
|-
| 311
|<math> \frac{1}{x}</math>
|<math> -i\pi\sgn(\xi)</math>
|<math> -i\sqrt{\frac{\pi}{2}}\sgn(\omega)</math>
|<math> -i\pi\sgn(\omega)</math>
|Here {{math|sgn(''ξ'')}} is the [[sign function]]. Note that {{math|{{sfrac|1|''x''}}}} is not a distribution. It is necessary to use the [[Cauchy principal value]] when testing against [[Schwartz functions]]. This rule is useful in studying the [[Hilbert transform]].
|-
| 312
|<math>\begin{align}
&\frac{1}{x^n} \\
&:= \frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log |x|
\end{align}</math>
|<math> -i\pi \frac{(-i 2\pi \xi)^{n-1}}{(n-1)!} \sgn(\xi)</math>
|<math> -i\sqrt{\frac{\pi}{2}}\, \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>
|<math> -i\pi \frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)</math>
|{{math|{{sfrac|1|''x''<sup>''n''</sup>}}}} is the [[homogeneous distribution]] defined by the distributional derivative{{br}}<math>\frac{(-1)^{n-1}}{(n-1)!}\frac{d^n}{dx^n}\log|x|</math>
|-
| 313
|<math> |x|^\alpha</math>
|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|2\pi\xi|^{\alpha+1}}</math>
|<math> \frac{-2}{\sqrt{2\pi}}\, \frac{\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} </math>
|<math> -\frac{2\sin\left(\frac{\pi\alpha}{2}\right)\Gamma(\alpha+1)}{|\omega|^{\alpha+1}} </math>
|This formula is valid for {{math|0 > ''α'' > −1}}. For {{math|''α'' > 0}} some singular terms arise at the origin that can be found by differentiating 320. If {{math|Re ''α'' > −1}}, then {{math|{{abs|''x''}}<sup>''α''</sup>}} is a locally integrable function, and so a tempered distribution. The function {{math|''α'' ↦ {{abs|''x''}}<sup>''α''</sup>}} is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted {{math|{{abs|''x''}}<sup>''α''</sup>}} for {{math|''α'' ≠ −1, −3, ...}} (See [[homogeneous distribution]].)
|-
| <!-- Should we call it 313a ? Doesn't necessarily need a number, because it is a special case. -->
|<math> \frac{1}{\sqrt{|x|}} </math>
|<math> \frac{1}{\sqrt{|\xi|}} </math>
|<math> \frac{1}{\sqrt{|\omega|}}</math>
|<math> \frac{\sqrt{2\pi}}{\sqrt{|\omega|}} </math>
| Special case of 313.
|-
| 314
|<math> \sgn(x)</math>
|<math> \frac{1}{i\pi \xi}</math>
|<math> \sqrt{\frac{2}{\pi}} \frac{1}{i\omega } </math>
|<math> \frac{2}{i\omega }</math>
|The dual of rule 311. This time the Fourier transforms need to be considered as a [[Cauchy principal value]].
|-
| 315
|<math> u(x)</math>
|<math> \frac{1}{2}\left(\frac{1}{i \pi \xi} + \delta(\xi)\right)</math>
|<math> \sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>
|<math> \pi\left( \frac{1}{i \pi \omega} + \delta(\omega)\right)</math>
|The function {{math|''u''(''x'')}} is the Heaviside [[Heaviside step function|unit step function]]; this follows from rules 101, 301, and 314.
|-
| 316
|<math> \sum_{n=-\infty}^{\infty} \delta (x - n T)</math>
|<math> \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( \xi -\frac{k }{T}\right)</math>
|<math> \frac{\sqrt{2\pi }}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>
|<math> \frac{2\pi}{T}\sum_{k=-\infty}^{\infty} \delta \left( \omega -\frac{2\pi k}{T}\right)</math>
|This function is known as the [[Dirac comb]] function. This result can be derived from 302 and 102, together with the fact that{{br}}<math>\begin{align}
& \sum_{n=-\infty}^{\infty} e^{inx} \\
= {}& 2\pi\sum_{k=-\infty}^{\infty} \delta(x+2\pi k)
\end{align}</math>{{br}}as distributions.
|-
| 317
|<math> J_0 (x)</math>
|<math> \frac{2\, \operatorname{rect}(\pi\xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>
|<math> \sqrt{\frac{2}{\pi}} \, \frac{\operatorname{rect}\left(  \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2\,\operatorname{rect}\left(\frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}}</math>
| The function {{math|''J''<sub>0</sub>(''x'')}} is the zeroth order [[Bessel function]] of first kind.
|-
| 318
|<math> J_n (x)</math>
|<math> \frac{2 (-i)^n T_n (2 \pi \xi) \operatorname{rect}(\pi \xi)}{\sqrt{1 - 4 \pi^2 \xi^2}} </math>
|<math> \sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2(-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
| This is a generalization of 317. The function {{math|''J<sub>n</sub>''(''x'')}} is the {{mvar|n}}th order [[Bessel function]] of first kind. The function {{math|''T<sub>n</sub>''(''x'')}} is the [[Chebyshev polynomials|Chebyshev polynomial of the first kind]].
|-
| 319
|<math> \log \left| x \right|</math>
|<math> -\frac{1}{2} \frac{1}{\left| \xi \right|} - \gamma \delta \left( \xi \right) </math>
|<math> -\frac{\sqrt\frac{\pi}{2}}{\left| \omega \right|} - \sqrt{2 \pi} \gamma \delta \left( \omega \right) </math>
|<math> -\frac{\pi}{\left| \omega \right|} - 2 \pi \gamma \delta \left( \omega \right) </math>
|{{mvar|γ}} is the [[Euler–Mascheroni constant]]. It is necessary to use a finite part integral when testing {{math|{{sfrac|1|{{abs|''ξ''}}}}}} or {{math|{{sfrac|1|{{abs|''ω''}}}}}}against [[Schwartz functions]]. The details of this might change the coefficient of the delta function.
|- 
| 320
|<math> \left( \mp ix \right)^{-\alpha}</math>
|<math> \frac{\left(2\pi\right)^\alpha}{\Gamma\left(\alpha\right)}u\left(\pm \xi \right)\left(\pm \xi \right)^{\alpha-1} </math>
|<math> \frac{\sqrt{2\pi}}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>
|<math> \frac{2\pi}{\Gamma\left(\alpha\right)}u\left(\pm\omega\right)\left(\pm\omega\right)^{\alpha-1} </math>
|This formula is valid for {{math|1 > ''α'' > 0}}. Use differentiation to derive formula for higher exponents. {{mvar|u}} is the Heaviside function.
|}

=== Two-dimensional functions ===
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|400
|<math> f(x,y)</math>
|<math>\begin{align}& \hat{f}(\xi_x, \xi_y)\triangleq \\ & \iint f(x,y) e^{-i 2\pi(\xi_x x+\xi_y y)}\,dx\,dy \end{align}</math>
|<math>\begin{align}& \hat{f}(\omega_x,\omega_y)\triangleq \\ & \frac{1}{2 \pi} \iint f(x,y) e^{-i (\omega_x x +\omega_y y)}\, dx\,dy  \end{align}</math>
|<math>\begin{align}& \hat{f}(\omega_x,\omega_y)\triangleq \\ & \iint f(x,y) e^{-i(\omega_x x+\omega_y y)}\, dx\,dy  \end{align}</math>
|The variables {{mvar|ξ<sub>x</sub>}}, {{mvar|ξ<sub>y</sub>}}, {{mvar|ω<sub>x</sub>}}, {{mvar|ω<sub>y</sub>}} are real numbers. The integrals are taken over the entire plane.
|-
|401
|<math> e^{-\pi\left(a^2x^2+b^2y^2\right)}</math>
|<math> \frac{1}{|ab|} e^{-\pi\left(\frac{\xi_x^2}{a^2} + \frac{\xi_y^2}{b^2}\right)}</math>
|<math> \frac{1}{2\pi\,|ab|} e^{-\frac{1}{4\pi}\left(\frac{\omega_x^2}{a^2} + \frac{\omega_y^2}{b^2}\right)}</math>
|<math> \frac{1}{|ab|} e^{-\frac{1}{4\pi}\left(\frac{\omega_x^2}{a^2} + \frac{\omega_y^2}{b^2}\right)}</math>
|Both functions are Gaussians, which may not have unit volume.
|-
|402
|<math> \operatorname{circ}\left(\sqrt{x^2+y^2}\right)</math>
|<math> \frac{J_1\left(2 \pi \sqrt{\xi_x^2+\xi_y^2}\right)}{\sqrt{\xi_x^2+\xi_y^2}}</math>
|<math> \frac{J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}</math>
|<math> \frac{2\pi J_1\left(\sqrt{\omega_x^2+\omega_y^2}\right)}{\sqrt{\omega_x^2+\omega_y^2}}</math>
|The function is defined by {{math|1=circ(''r'') = 1}} for {{math|0 ≤ ''r'' ≤ 1}}, and is 0 otherwise. The result is the amplitude distribution of the [[Airy disk]], and is expressed using {{math|''J''<sub>1</sub>}} (the order-1 [[Bessel function]] of the first kind).<ref>{{harvnb|Stein|Weiss|1971|loc=Thm. IV.3.3}}</ref>
|-
|403
|<math> \frac{1}{\sqrt{x^2+y^2}}</math>
|<math> \frac{1}{\sqrt{\xi_x^2+\xi_y^2}}</math>
|<math> \frac{1}{\sqrt{\omega_x^2+\omega_y^2}}</math>
|<math> \frac{2\pi}{\sqrt{\omega_x^2+\omega_y^2}}</math>
|This is the [[Hankel transform]] of {{math|1=''r''<sup>−1</sup>}}, a 2-D Fourier "self-transform".<ref>{{harvnb|Easton|2010}}</ref>
|-
|404
|<math> \frac{i}{x+i y}</math>
|<math> \frac{1}{\xi_x+i\xi_y}</math>
|<math> \frac{1}{\omega_x+i\omega_y}</math>
|<math> \frac{2\pi}{\omega_x+i\omega_y}</math>
|<!--This formula was used in constructing the ground state wavefunction of two-dimensional <math> p_x+ip_y</math> superconductors<ref>Phys. Rev. B 97 (10), 104501 (2018)</ref>-->
|}

===Formulas for general {{math|''n''}}-dimensional functions===
{| class="wikitable"
! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks
|-
|500
|<math> f(\mathbf x)\,</math>
|<math>\begin{align} &\hat{f_1}(\boldsymbol \xi) \triangleq \\ &\int_{\mathbb{R}^n}f(\mathbf x) e^{-i 2\pi \boldsymbol \xi \cdot \mathbf x }\, d \mathbf x \end{align}</math>
|<math>\begin{align} &\hat{f_2}(\boldsymbol \omega) \triangleq \\ &\frac{1}{{(2 \pi)}^\frac{n}{2}} \int_{\mathbb{R}^n} f(\mathbf x) e^{-i \boldsymbol \omega \cdot \mathbf x}\, d \mathbf x \end{align}</math>
|<math>\begin{align} &\hat{f_3}(\boldsymbol \omega) \triangleq \\ &\int_{\mathbb{R}^n}f(\mathbf x) e^{-i \boldsymbol \omega \cdot \mathbf x}\, d \mathbf x \end{align}</math>
|
|-
|501
|<math> \chi_{[0,1]}(|\mathbf x|)\left(1-|\mathbf x|^2\right)^\delta</math>
|<math> \frac{\Gamma(\delta+1)}{\pi^\delta\,|\boldsymbol \xi|^{\frac{n}{2} + \delta}}  J_{\frac{n}{2}+\delta}(2\pi|\boldsymbol \xi|)</math>
|<math> 2^\delta \, \frac{\Gamma(\delta+1)}{\left|\boldsymbol \omega\right|^{\frac{n}{2}+\delta}}  J_{\frac{n}{2}+\delta}(|\boldsymbol \omega|)</math>
|<math> \frac{\Gamma(\delta+1)}{\pi^\delta} \left|\frac{\boldsymbol \omega}{2\pi}\right|^{-\frac{n}{2}-\delta} J_{\frac{n}{2}+\delta}(\!|\boldsymbol \omega|\!)</math>
|The function {{math|''χ''<sub>[0, 1]</sub>}} is the [[indicator function]] of the interval {{math|[0, 1]}}. The function {{math|Γ(''x'')}} is the gamma function. The function {{math|''J''<sub>{{sfrac|''n''|2}} + ''δ''</sub>}} is a Bessel function of the first kind, with order {{math|{{sfrac|''n''|2}} + ''δ''}}. Taking {{math|1=''n'' = 2}} and {{math|1=''δ'' = 0}} produces 402.<ref>{{harvnb|Stein|Weiss|1971|loc=Thm. 4.15}}</ref>
|-
|502
|<math> |\mathbf x|^{-\alpha}, \quad 0 < \operatorname{Re} \alpha < n.</math>
|<math> \frac{(2\pi)^{\alpha}}{c_{n, \alpha}} |\boldsymbol \xi|^{-(n - \alpha)}</math>
|<math> \frac{(2\pi)^{\frac{n}{2}}}{c_{n, \alpha}} |\boldsymbol \omega|^{-(n - \alpha)}</math>
|<math> \frac{(2\pi)^{n}}{c_{n, \alpha}} |\boldsymbol \omega|^{-(n - \alpha)}</math>
|See [[Riesz potential]] where the constant is given by{{br}}<math>c_{n, \alpha} = \pi^\frac{n}{2} 2^\alpha \frac{\Gamma\left(\frac{\alpha}{2}\right)}{\Gamma\left(\frac{n - \alpha}{2}\right)}.</math>{{br}}The formula also holds for all {{math|''α'' ≠ ''n'', ''n'' + 2, ...}} by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See [[homogeneous distribution]].<ref group=note>In {{harvnb|Gelfand|Shilov|1964|p=363}}, with the non-unitary conventions of this table, the transform of <math>|\mathbf x|^\lambda</math> is given to be{{br}} <math>2^{\lambda+n}\pi^{\tfrac12 n}\frac{\Gamma\left(\frac{\lambda+n}{2}\right)}{\Gamma\left(-\frac{\lambda}{2}\right)}|\boldsymbol\omega|^{-\lambda-n}</math>{{br}}from which this follows, with <math>\lambda=-\alpha</math>.</ref>
|-
|503
|<math> \frac{1}{\left|\boldsymbol \sigma\right|\left(2\pi\right)^\frac{n}{2}} e^{-\frac{1}{2} \mathbf x^{\mathrm T} \boldsymbol \sigma^{-\mathrm T} \boldsymbol \sigma^{-1} \mathbf x}</math>
|<math> e^{-2\pi^2 \boldsymbol \xi^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \xi} </math>
|<math> (2\pi)^{-\frac{n}{2}} e^{-\frac{1}{2} \boldsymbol \omega^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \omega} </math>
|<math> e^{-\frac{1}{2} \boldsymbol \omega^{\mathrm T} \boldsymbol \sigma \boldsymbol \sigma^{\mathrm T} \boldsymbol \omega} </math>
|This is the formula for a [[multivariate normal distribution]] normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page, {{math|'''Σ''' {{=}} '''σ''' '''σ'''<sup>T</sup>}} and {{math|'''Σ'''<sup>−1</sup> {{=}} '''σ'''<sup>−T</sup> '''σ'''<sup>−1</sup>}}
|-
|504
|<math> e^{-2\pi\alpha|\mathbf x|}</math>
| <math>\frac{c_n\alpha}{\left(\alpha^2+|\boldsymbol{\xi}|^2\right)^\frac{n+1}{2}}</math>
|<math>\frac{c_n (2\pi)^{\frac{n+2}{2}} \alpha}{\left(4\pi^2\alpha^2+|\boldsymbol{\omega}|^2\right)^\frac{n+1}{2}}</math>
|<math>\frac{c_n (2\pi)^{n+1} \alpha}{\left(4\pi^2\alpha^2+|\boldsymbol{\omega}|^2\right)^\frac{n+1}{2}}</math>
|Here<ref>{{harvnb|Stein|Weiss|1971|p=6}}</ref>{{br}}<math>c_n=\frac{\Gamma\left(\frac{n+1}{2}\right)}{\pi^\frac{n+1}{2}},</math> {{math|Re(''α'') > 0}}
|}