# Hankel变换

${\displaystyle F_{\nu }(k)=\int _{0}^{\infty }f(r)J_{\nu }(kr)rdr}$

${\displaystyle f(r)=\int _{0}^{\infty }F_{\nu }(k)J_{\nu }(kr)kdk}$

## 正交性

${\displaystyle \int _{0}^{\infty }J_{\nu }(kr)J_{\nu }(k'r)r~\operatorname {d} r={\frac {\delta (k-k')}{k}}}$

## 与其他函数变换的关系

### 傅立叶变换

${\displaystyle F({\boldsymbol {k}})=\iint f({\boldsymbol {r}})e^{i{\boldsymbol {k}}\cdot {\boldsymbol {r}}}d{\boldsymbol {r}}}$

${\displaystyle F({\boldsymbol {k}})=\int _{0}^{\infty }\int _{0}^{2\pi }f(r,\theta )e^{ikr\cos \theta }rdrd\theta }$

${\displaystyle F({\boldsymbol {k}})=F(k)=2\pi \int _{0}^{\infty }f(r)J_{0}(kr)rdr}$

## 常见汉克尔变换函数对

${\displaystyle f(r)\,}$  ${\displaystyle F_{0}(k)\,}$
${\displaystyle 1\,}$  ${\displaystyle \delta (k)/k\,}$
${\displaystyle 1/r\,}$  ${\displaystyle 1/k\,}$
${\displaystyle r\,}$  ${\displaystyle -1/k^{3}\,}$
${\displaystyle r^{3}\,}$  ${\displaystyle 9/k^{5}\,}$
${\displaystyle r^{m}\,}$  ${\displaystyle {\frac {2^{m+1}\Gamma (m/2+1)}{k^{m+2}\Gamma (-m/2)}}\,}$  for -2<Re(m)<-1/2
${\displaystyle {\frac {1}{\sqrt {r^{2}+z^{2}}}}\,}$  ${\displaystyle {\frac {e^{-k|z|}}{k}}={\sqrt {\frac {2|z|}{\pi k}}}K_{-1/2}(k|z|)\,}$
${\displaystyle {\frac {1}{r^{2}+z^{2}}}\,}$  ${\displaystyle K_{0}(kz)\,}$ , ${\displaystyle z}$ 可为复数
${\displaystyle e^{iar}/r\,}$  ${\displaystyle i/{\sqrt {a^{2}-k^{2}}}\quad (a>0,k
${\displaystyle \,}$  ${\displaystyle 1/{\sqrt {k^{2}-a^{2}}}\quad (a>0,k>a)\,}$
${\displaystyle e^{-a^{2}r^{2}/2}\,}$  ${\displaystyle {\frac {e^{-k^{2}/2a^{2}}}{a^{2}}}}$
${\displaystyle -r^{2}f(r)\,}$  ${\displaystyle {\frac {\operatorname {d} ^{2}F_{0}}{\operatorname {d} k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d} F_{0}}{\operatorname {d} k}}}$
${\displaystyle f(r)\,}$  ${\displaystyle F_{\nu }(k)\,}$
${\displaystyle r^{s}\,}$  ${\displaystyle {\frac {\Gamma \left({\frac {1}{2}}(2+\nu +s)\right)}{\Gamma ({\tfrac {1}{2}}(\nu -s))}}{\frac {2^{s+1}}{k^{s+2}}}\,}$
${\displaystyle r^{\nu -2s}\Gamma \left(s,r^{2}h\right)\,}$  ${\displaystyle {\frac {1}{2}}\left({\frac {k}{2}}\right)^{2s-\nu -2}\gamma \left(1-s+\nu ,{\frac {k^{2}}{4h}}\right)\,}$
${\displaystyle e^{-r^{2}}r^{\nu }U\left(a,b,r^{2}\right)\,}$  ${\displaystyle {\frac {\Gamma (2+\nu -b)}{2\Gamma (2+\nu -b+a)}}\left({\frac {k}{2}}\right)^{\nu }e^{-{\frac {k^{2}}{4}}}\,_{1}F_{1}\left(a,2+a-b+\nu ,{\frac {k^{2}}{4}}\right)}$
${\displaystyle -r^{2}f(r)\,}$  ${\displaystyle {\frac {\operatorname {d} ^{2}F_{\nu }}{\operatorname {d} k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d} F_{\nu }}{\operatorname {d} k}}-{\frac {\nu ^{2}}{k^{2}}}F_{\nu }}$