Editing Fourier transform (section)

===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}}
|}