Editing Integral transform (section)

==Table of transforms==
{| class="wikitable"
|+ Table of integral transforms
|-
! scope="col" | Transform
! scope="col" | Symbol
! scope="col" | ''K''
! scope="col" | ''f''(''t'')
! scope="col" | ''t''<sub>1</sub>
! scope="col" | ''t''<sub>2</sub>
! scope="col" | ''K''<sup>−1</sup>
! scope="col" | ''u''<sub>1</sub>
! scope="col" | ''u''<sub>2</sub>
|-
| [[Abel transform]]
| F, f
| <math>\frac{2t}{\sqrt{t^2-u^2}}</math>
|
| <math>u</math>
| <math>\infty</math>
| <math>\frac{-1}{\pi\sqrt{u^2\!-\!t^2}}\frac{d}{du}</math> <ref> Assuming the Abel transform is not discontinuous at <math>u</math>.</ref>
| ''t''
| <math>\infty</math>
|-
| Associated Legendre transform
| <math>\mathcal{J}_{n,m}</math>
| <math>(1-x^2)^{-m/2}P^{m}_n(x)</math>
|
| <math>-1</math>
| <math>1</math>
|
| <math>0</math>
| <math>\infty</math>
|-
| [[Fourier transform]]
| <math>\mathcal{F}</math>
| <math>e^{-2\pi iut}</math>
| <math>L_1</math>
| <math>-\infty</math>
| <math>\infty</math>
| <math>e^{2\pi iut}</math>
| <math>-\infty</math>
| <math>\infty</math>
|-
| [[Fourier sine transform]]
| <math>\mathcal{F}_s</math>
| <math>\sqrt{\frac{2}{\pi}} \sin(ut)</math>
| on <math>[0,\infty)</math>, real-valued
| <math>0</math>
| <math>\infty</math>
| <math>\sqrt{\frac{2}{\pi}} \sin(ut)</math>
| <math>0</math>
| <math>\infty</math>
|-
| [[Fourier cosine transform]]
| <math>\mathcal{F}_c</math>
| <math>\sqrt{\frac{2}{\pi}} \cos(ut)</math>
| on <math>[0,\infty)</math>, real-valued
| <math>0</math>
| <math>\infty</math>
| <math>\sqrt{\frac{2}{\pi}} \cos(ut)</math>
| <math>0</math>
| <math>\infty</math>
|-
| [[Hankel transform]]
| 
| <math>t\,J_\nu(ut)</math>
|
| <math>0</math>
| <math>\infty</math>
| <math>u\,J_\nu(ut)</math>
| <math>0</math>
| <math>\infty</math>
|-
| [[Hartley transform]]
| <math>\mathcal{H}</math>
| <math>\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| <math>\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}</math>
| <math>-\infty</math>
| <math>\infty</math>
|-
| [[Hermite transform]]
| <math>H</math>
| <math>e^{-x^2} H_n(x)</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| 
| <math>0</math>
| <math>\infty</math>
|-
| [[Hilbert transform]]
| <math>\mathcal{H}il</math>
| <math>\frac{1}{\pi}\frac{1}{u-t}</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| <math>\frac{1}{\pi}\frac{1}{u-t}</math>
| <math>-\infty</math>
| <math>\infty</math>
|-
| [[Jacobi transform]]
| <math>J</math>
| <math>(1-x)^\alpha\  (1+x)^\beta \ P_n^{\alpha,\beta}(x)</math>
|
| <math>-1</math>
| <math>1</math>
| 
| <math>0</math>
| <math>\infty</math>
|-
| [[Laguerre transform]]
| <math>L</math>
| <math>e^{-x}\  x^\alpha \ L_n^{\alpha}(x)</math>
|
| <math>0</math>
| <math>\infty</math>
| 
| <math>0</math>
| <math>\infty</math>
|-
| [[Laplace transform]]
| <math>\mathcal{L}</math>
| <math>e^{-ut}</math>
|
| <math>0</math>
| <math>\infty</math>
| <math>\frac{e^{ut}}{2\pi i}</math>
| <math>c\!-\!i\infty</math>
| <math>c\!+\!i\infty</math>
|-
| [[Legendre transform (integral transform)|Legendre transform]]
| <math>\mathcal{J}</math>
| <math>P_n(x)\,</math>
|
| <math>-1</math>
| <math>1</math>
|
| <math>0</math>
| <math>\infty</math>
|-
| [[Mellin transform]]
| <math>\mathcal{M}</math>
| <math>t^{u-1}</math>
|
| <math>0</math>
| <math>\infty</math>
| <math>\frac{t^{-u}}{2\pi i}\,</math><ref> Some conditions apply, see [[Mellin inversion theorem]] for details. </ref>
| <math>c\!-\!i\infty</math>
| <math>c\!+\!i\infty</math>
|-
| [[Two-sided Laplace transform|Two-sided Laplace<br>transform]]
| <math>\mathcal{B}</math>
| <math>e^{-ut}</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| <math>\frac{e^{ut}}{2\pi i}</math>
| <math>c\!-\!i\infty</math>
| <math>c\!+\!i\infty</math>
|-
| [[Poisson kernel]]
|
| <math>\frac{1-r^2}{1-2r\cos\theta +r^2}</math>
|
| <math>0</math>
| <math>2\pi</math>
| 
| 
| 
|-
| [[Radon transform]]
| Rƒ
| <math> \delta(x\cos\theta+y\sin\theta-t)</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| 
| 
| 
|-
| [[Weierstrass transform]]
| <math>\mathcal{W}</math>
| <math>\frac{e^{-\frac{(u-t)^2}{4}}}{\sqrt{4\pi}}\,</math>
|
| <math>-\infty</math>
| <math>\infty</math>
| <math>\frac{e^{\frac{(u-t)^2}{4}}}{i\sqrt{4\pi}}</math>
| <math>c\!-\!i\infty</math>
| <math>c\!+\!i\infty</math>
|-
| [[X-ray transform]]
| Xƒ
| 
|
| <math>-\infty</math>
| <math>\infty</math>
| 
| 
| 
|}
In the limits of integration for the inverse transform, ''c'' is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, ''c'' must be greater than the largest real part of the zeroes of the transform function.

Note that there are alternative notations and conventions for the Fourier transform.