Editing Fourier transform (section)

=== 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>-->
|}