# 积分变换

## 概述

$(Tf)(u) = \int \limits_{t_1}^{t_2} K(t, u)\, f(t)\, dt$

$f(t) = \int \limits_{u_1}^{u_2} K^{-1}( u,t )\, (Tf(u))\, du$

$K^{-1}( u,t )$ 稱為反核(inverse kernel)。

## 積分變換表列

en:Hartley transform $\mathcal{H}$ $\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}$ $-\infty\,$ $\infty\,$ $\frac{\cos(ut)+\sin(ut)}{\sqrt{2 \pi}}$ $-\infty\,$ $\infty\,$
en: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
$\mathcal{B}$ $e^{-ut}\,$ $-\infty\,$ $\infty\,$ $\frac{e^{+ut}}{2\pi i}$ $c\!-\!i\infty$ $c\!+\!i\infty$

en:Weierstrass transform $\mathcal{W}$ $\frac{e^{-(u-t)^2/4}}{\sqrt{4\pi}}\,$ $-\infty\,$ $\infty\,$ $\frac{e^{+(u-t)^2/4}}{i\sqrt{4\pi}}$ $c\!-\!i\infty$ $c\!+\!i\infty$
en:Hankel transform $t\,J_\nu(ut)$ $0\,$ $\infty\,$ $u\,J_\nu(ut)$ $0\,$ $\infty\,$
en:Abel transform $\frac{2t}{\sqrt{t^2-u^2}}$ $u\,$ $\infty\,$ $\frac{-1}{\pi\sqrt{u^2\!-\!t^2}}\frac{d}{du}$ $t\,$ $\infty\,$

en:Poisson kernel $\frac{1-r^2}{1-2r\cos\theta +r^2}$ $0\,$ $2\pi\,$
Identity transform $\delta (u-t)\,$ $t_1 $t_2>u\,$ $\delta (t-u)\,$ $u_1\!<\!t$ $u_2\!>\!t$