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⁻¹	u₁	u₂
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	$\mathcal{F}_s$	$\sqrt{\frac{2}{\pi}} \sin(ut)$	$[0,\infty )$	$0$	$\infty$	$\sqrt{\frac{2}{\pi}} \sin(ut)$	$0$	$\infty$
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	$H$	$e^{-x^{2}}H_{n}(x)$		$-\infty$	$\infty$		$0$	$\infty$
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	$J$	$(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)$		$-1$	$1$		$0$	$\infty$
Laguerre transform	$L$	$e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)$		$0$	$\infty$		$0$	$\infty$
Laplace transform	${\mathcal {L}}$	e^−ut		0	$\infty$	$\frac{e^{ut}}{2\pi i}$	$c\!-\!i\infty$	$c\!+\!i\infty$
Legendre transform	${\mathcal {J}}$	$P_{n}(x)\,$		$-1$	$1$		$0$	$\infty$
Mellin transform	${\mathcal {M}}$	t^u−1		0	$\infty$	$\frac{t^{-u}}{2\pi i}\,$	$c\!-\!i\infty$	$c\!+\!i\infty$
Two-sided Laplace transform	${\mathcal {B}}$	e^−ut		$-\infty$	$\infty$	$\frac{e^{ut}}{2\pi i}$	$c\!-\!i\infty$	$c\!+\!i\infty$
Poisson kernel		$\frac{1-r^2}{1-2r\cos\theta +r^2}$		0	2π
Radon Transform	Rƒ			$-\infty$	$\infty$
Weierstrass transform	${\mathcal {W}}$	$\frac{e^{-\frac{(u-t)^2}{4}}}{\sqrt{4\pi}}\,$		$-\infty$	$\infty$	$\frac{e^{\frac{(u-t)^2}{4}}}{i\sqrt{4\pi}}$	$c\!-\!i\infty$	$c\!+\!i\infty$