# 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

F, f ${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}}$  u ${\displaystyle \infty }$  ${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}}$  [1] t ${\displaystyle \infty }$

${\displaystyle {\mathcal {H}}}$  ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$  ${\displaystyle -\infty }$  ${\displaystyle \infty }$  ${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$  ${\displaystyle -\infty }$  ${\displaystyle \infty }$
${\displaystyle H}$  ${\displaystyle e^{-x^{2}}H_{n}(x)}$  ${\displaystyle -\infty }$  ${\displaystyle \infty }$  ${\displaystyle 0}$  ${\displaystyle \infty }$

${\displaystyle J}$  ${\displaystyle (1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)}$  ${\displaystyle -1}$  ${\displaystyle 1}$  ${\displaystyle 0}$  ${\displaystyle \infty }$
${\displaystyle L}$  ${\displaystyle e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)}$  ${\displaystyle 0}$  ${\displaystyle \infty }$  ${\displaystyle 0}$  ${\displaystyle \infty }$

${\displaystyle {\mathcal {J}}}$  ${\displaystyle P_{n}(x)\,}$  ${\displaystyle -1}$  ${\displaystyle 1}$  ${\displaystyle 0}$  ${\displaystyle \infty }$

${\displaystyle {\mathcal {W}}}$  ${\displaystyle {\frac {e^{-{\frac {(u-t)^{2}}{4}}}}{\sqrt {4\pi }}}\,}$  ${\displaystyle -\infty }$  ${\displaystyle \infty }$  ${\displaystyle {\frac {e^{\frac {(u-t)^{2}}{4}}}{i{\sqrt {4\pi }}}}}$  ${\displaystyle c\!-\!i\infty }$  ${\displaystyle c\!+\!i\infty }$
${\displaystyle -\infty }$  ${\displaystyle \infty }$

## 参见

1. ^ Assuming the Abel transform is not discontinuous at ${\displaystyle u}$ .
2. ^ Some conditions apply, see Mellin inversion theorem for details.