漢克爾變換

漢克爾變換是指對任何給定函數 ${\displaystyle f(r)}$ 以第一類貝塞爾函數 ${\displaystyle J_{\nu }(kr)}$ 作無窮級數展開，貝塞爾函數 ${\displaystyle J_{\nu }(kr)}$ 的階數不變，級數各項 ${\displaystyle k}$ 作變化。各項 ${\displaystyle J_{\nu }(kr)}$ 前係數 ${\displaystyle F_{\nu }}$ 構成了變換函數。對於函數 ${\displaystyle f(r)}$, 其 ${\displaystyle \nu }$ 階貝塞爾函數的漢克爾變換（${\displaystyle k}$ 為自變量）為

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

其中，${\displaystyle J_{\nu }}$ 為階數為 ${\displaystyle \nu }$ 的第一類貝塞爾函數，${\displaystyle \nu \geq -1/2}$。對應的，逆漢克爾變換 ${\displaystyle F_{\nu }(k)}$ 定義為

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

漢克爾變換是一種積分變換，最早由德國數學家赫爾曼·漢克爾提出，又被稱為傅立葉-貝塞爾變換。

正交性

貝塞爾函數構成 正交函數族 權重因子為 r:

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

其中 ${\displaystyle k}$ 與 ${\displaystyle k'}$ 大於零。

與其他函數變換的關係

傅立葉變換

零階漢克爾函數即為圓對稱函數的二維傅立葉變換。給定二維函數 ${\displaystyle F({\boldsymbol {r}})}$ ，徑向矢量為 ${\displaystyle {\boldsymbol {r}}}$，其傅立葉變換為

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

不失一般性，選擇極坐標 ${\displaystyle (r,\theta )}$ ，使得矢量 ${\displaystyle {\boldsymbol {k}}}$ 方向指向 ${\displaystyle \theta =0}$ 。極坐標下的傅立葉變換寫作

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

其中 ${\displaystyle \theta }$ 為矢量 ${\displaystyle {\boldsymbol {k}}}$ 與 ${\displaystyle {\boldsymbol {r}}}$ 間夾角。如果函數 ${\displaystyle f}$ 恰為圓對稱不依賴角變量 ${\displaystyle \theta }$ ，${\displaystyle f\equiv f(r)}$ ，對角度 ${\displaystyle \theta }$ 的積分可以提出，傅立葉變換寫作

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

此式恰為 ${\displaystyle f(r)}$ 的零階漢克爾變換的 ${\displaystyle 2\pi }$ 倍。

常見漢克爾變換函數對

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

參見條目

• 傅里葉變換