Integral Transforms on Wikipedia

Key Integral Transforms

An extension of the Fourier transforms and a necessary part of the earliest universe and the definition of functions throughout the 202 notations, this chart comes directly from the Wikipedia page and is posted here for easy and quick reference. –BEC

Transform Symbol K f(t) t1 t2 K−1u1u2
Abel transform \frac{2t}{\sqrt{t^2-u^2}}u \infty \frac{-1}{\pi\sqrt{u^2\!-\!t^2}}\frac{d}{du}\infty
Fourier transform{\mathcal {F}} e^{-2\pi iut} L_{1}-\infty \infty e^{2\pi iut}-\infty \infty
Fourier sine transform \mathcal{F}_s\sqrt{\frac{2}{\pi}} \sin(ut)[0,\infty ) {\displaystyle 0} \infty \sqrt{\frac{2}{\pi}} \sin(ut) {\displaystyle 0}\infty
Fourier cosine transform \mathcal{F}_c\sqrt{\frac{2}{\pi}} \cos(ut)[0,\infty )0\infty \sqrt{\frac{2}{\pi}} \cos(ut)0\infty
Hankel transform t\,J_\nu(ut)0\infty u\,J_\nu(ut)0 \infty
Hartley transform {\mathcal {H}}\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}} -\infty \infty \frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}-\infty \infty
Hermite transform H {\displaystyle e^{-x^{2}}H_{n}(x)}-\infty \infty {\displaystyle 0} \infty
Hilbert transform \mathcal{H}il \frac{1}{\pi}\frac{1}{u-t}-\infty \infty \frac{1}{\pi}\frac{1}{u-t} -\infty \infty
Jacobi transform J{\displaystyle (1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)}-1 1 {\displaystyle 0} \infty
Laguerre transformL{\displaystyle e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)} {\displaystyle 0} \infty {\displaystyle 0}\infty
Laplace transform{\mathcal {L}}e−ut0 \infty \frac{e^{ut}}{2\pi i}c\!-\!i\inftyc\!+\!i\infty
Legendre transform {\mathcal {J}}P_{n}(x)\, -1 1 {\displaystyle 0} \infty
Mellin transform{\mathcal {M}}tu−10\infty \frac{t^{-u}}{2\pi i}\, c\!-\!i\infty c\!+\!i\infty
Two-sided Laplace
transform
{\mathcal {B}}e−ut-\infty \infty \frac{e^{ut}}{2\pi i}c\!-\!i\inftyc\!+\!i\infty
Poisson kernel\frac{1-r^2}{1-2r\cos\theta +r^2}0
Radon Transform -\infty \infty
Weierstrass transform {\mathcal {W}} \frac{e^{-\frac{(u-t)^2}{4}}}{\sqrt{4\pi}}\, -\infty \infty \frac{e^{\frac{(u-t)^2}{4}}}{i\sqrt{4\pi}}c\!-\!i\inftyc\!+\!i\infty

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