漢克爾變換

汉克尔变换是指对任何给定函数 ${\displaystyle f(r)}$ 以第一类贝塞尔函数 ${\displaystyle J_{\nu }(kr)}$ 作无穷级数展开，贝塞尔函数 ${\displaystyle J_{\nu }(kr)}$ 的阶数不变，级数各项 ${\displaystyle k}$ 作变化。各项 ${\displaystyle J_{\nu }(kr)}$ 前系数 ${\displaystyle F_{\nu }}$ 构成了变换函数。对于函数 ${\displaystyle f(r)}$, 其 ${\displaystyle \nu }$ 阶贝塞尔函数的汉克尔变换（${\displaystyle k}$ 为自变量）为

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

其中，${\displaystyle J_{\nu }}$ 为阶数为 ${\displaystyle \nu }$ 的第一类贝塞尔函数，${\displaystyle \nu \geq -1/2}$。对应的，逆汉克尔变换 ${\displaystyle F_{\nu }(k)}$ 定义为

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

汉克尔变换是一种积分变换，最早由德国数学家赫尔曼·汉克尔提出，又被称为傅立叶-贝塞尔变换。

正交性

贝塞尔函数构成 正交函数族 权重因子为 r:

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

其中 ${\displaystyle k}$ 与 ${\displaystyle k'}$ 大于零。

与其他函数变换的关系

傅立叶变换

零阶汉克尔函数即为圆对称函数的二维傅立叶变换。给定二维函数 ${\displaystyle F({\boldsymbol {r}})}$ ，径向矢量为 ${\displaystyle {\boldsymbol {r}}}$，其傅立叶变换为

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

不失一般性，选择极坐标 ${\displaystyle (r,\theta )}$ ，使得矢量 ${\displaystyle {\boldsymbol {k}}}$ 方向指向 ${\displaystyle \theta =0}$ 。极坐标下的傅立叶变换写作

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

其中 ${\displaystyle \theta }$ 为矢量 ${\displaystyle {\boldsymbol {k}}}$ 与 ${\displaystyle {\boldsymbol {r}}}$ 间夹角。如果函数 ${\displaystyle f}$ 恰为圆对称不依赖角变量 ${\displaystyle \theta }$ ，${\displaystyle f\equiv f(r)}$ ，对角度 ${\displaystyle \theta }$ 的积分可以提出，傅立叶变换写作

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

此式恰为 ${\displaystyle f(r)}$ 的零阶汉克尔变换的 ${\displaystyle 2\pi }$ 倍。

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
${\displaystyle e^{iar}/r\,}$	${\displaystyle i/{\sqrt {a^{2}-k^{2}}}\quad (a>0,k<a)\,}$
${\displaystyle \,}$	${\displaystyle 1/{\sqrt {k^{2}-a^{2}}}\quad (a>0,k>a)\,}$
${\displaystyle e^{-a^{2}r^{2}/2}\,}$	${\displaystyle {\frac {e^{-k^{2}/2a^{2}}}{a^{2}}}}$
${\displaystyle -r^{2}f(r)\,}$	${\displaystyle {\frac {\operatorname {d} ^{2}F_{0}}{\operatorname {d} k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d} F_{0}}{\operatorname {d} k}}}$
${\displaystyle f(r)\,}$	${\displaystyle F_{\nu }(k)\,}$
${\displaystyle r^{s}\,}$	${\displaystyle {\frac {\Gamma \left({\frac {1}{2}}(2+\nu +s)\right)}{\Gamma ({\tfrac {1}{2}}(\nu -s))}}{\frac {2^{s+1}}{k^{s+2}}}\,}$
${\displaystyle r^{\nu -2s}\Gamma \left(s,r^{2}h\right)\,}$	${\displaystyle {\frac {1}{2}}\left({\frac {k}{2}}\right)^{2s-\nu -2}\gamma \left(1-s+\nu ,{\frac {k^{2}}{4h}}\right)\,}$
${\displaystyle e^{-r^{2}}r^{\nu }U\left(a,b,r^{2}\right)\,}$	${\displaystyle {\frac {\Gamma (2+\nu -b)}{2\Gamma (2+\nu -b+a)}}\left({\frac {k}{2}}\right)^{\nu }e^{-{\frac {k^{2}}{4}}}\,_{1}F_{1}\left(a,2+a-b+\nu ,{\frac {k^{2}}{4}}\right)}$
${\displaystyle -r^{2}f(r)\,}$	${\displaystyle {\frac {\operatorname {d} ^{2}F_{\nu }}{\operatorname {d} k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d} F_{\nu }}{\operatorname {d} k}}-{\frac {\nu ^{2}}{k^{2}}}F_{\nu }}$

参见条目

• 傅里叶变换