# 积分变换

## 概述

${\displaystyle (Tf)(u)=\int \limits _{t_{1}}^{t_{2}}K(t,u)\,f(t)\,dt}$

${\displaystyle f(t)=\int \limits _{u_{1}}^{u_{2}}K^{-1}(u,t)\,(Tf(u))\,du}$

${\displaystyle K^{-1}(u,t)}$  稱為反核（inverse kernel）。

## 積分變換表

${\displaystyle {\mathcal {H}}}$  ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$  ${\displaystyle -\infty \,}$  ${\displaystyle \infty \,}$  ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$  ${\displaystyle -\infty \,}$  ${\displaystyle \infty \,}$
Mellin变换 ${\displaystyle {\mathcal {M}}}$  ${\displaystyle t^{u-1}\,}$  ${\displaystyle 0\,}$  ${\displaystyle \infty \,}$  ${\displaystyle {\frac {t^{-u}}{2\pi i}}\,}$  ${\displaystyle c\!-\!i\infty }$  ${\displaystyle c\!+\!i\infty }$

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