# 柱諧函數

${\displaystyle V_{n,k}(\rho ,\varphi ,z)=P_{n,k}(\rho )\Phi _{n}(\varphi )Z_{k}(z)\,}$

## 定義

### 本徵方程的推導

${\displaystyle \nabla ^{2}V={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial V}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}V}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0}$

${\displaystyle V=P(\rho )\,\Phi (\varphi )\,Z(z)}$

${\displaystyle {\frac {\Phi Z}{\rho }}{\frac {d}{d\rho }}(\rho {\frac {dP}{d\rho }})+{\frac {PZ}{\rho ^{2}}}{\frac {d^{2}\Phi }{d\varphi ^{2}}}+P\Phi {\frac {d^{2}Z}{dz^{2}}}=0}$

${\displaystyle {\begin{cases}{\frac {1}{\rho }}{\frac {d}{d\rho }}(\rho {\frac {dP}{d\rho }})+{\frac {1}{\rho ^{2}}}\,(-n^{2})+k^{2}=0\\{\frac {1}{\Phi }}(d^{2}\Phi /d\varphi ^{2})=-n^{2}\\{\frac {1}{Z}}(d^{2}Z/dz^{2})=k^{2}\end{cases}}}$，整理得 ${\displaystyle {\begin{cases}\rho ^{2}P''+\rho P'+(k^{2}\rho ^{2}-n^{2})P=0\\\Phi ''+n^{2}\Phi =0\\Z''-k^{2}Z=0\end{cases}}}$

### 本徵方程的求解

${\displaystyle \Phi _{n}=\{\cos(n\varphi ),\sin(n\varphi )\}}$${\displaystyle n\in \mathbb {N} }$

${\displaystyle \Phi _{n}=\{e^{in\varphi },e^{-in\varphi }\}}$${\displaystyle n\in \mathbb {N} }$

${\displaystyle Z_{k}=\{\cosh(kz),\sinh(kz)\}}$

${\displaystyle Z_{k}=\{e^{kz},e^{-kz}\}}$

${\displaystyle Z_{k}=\{\cos(|k|z),\sin(|k|z)\}}$

${\displaystyle Z_{k}=\{e^{i|k|z},e^{-i|k|z}\}}$

${\displaystyle P}$的方程則是一個貝塞爾方程，它的解${\displaystyle P_{n,k}}$形式如下。

${\displaystyle k=0}$，則該方程簡化為一個歐拉方程

${\displaystyle P_{0,0}=\{1,\ln \rho \}}$
${\displaystyle P_{n,0}=\{\rho ^{n},\rho ^{-n}\},n\neq 0}$

${\displaystyle k}$是一個非零實數，則方程的解為第一類和/或第二類貝塞爾函數

${\displaystyle P_{n,k}=\{J_{n}(k\rho ),Y_{n}(k\rho )\}}$

${\displaystyle P_{n,k}=\{I_{n}(|k|\rho ),K_{n}(|k|\rho )\}}$

### 正交完備性

${\displaystyle \int _{0}^{\infty }d\rho \int _{0}^{2\pi }d\varphi \int _{-\infty }^{\infty }dz\left[V_{n,k}(\rho ,\theta ,\varphi )V_{n',k'}(\rho ,\theta ,\varphi )\right]={\frac {1}{C_{n,k}^{2}}}\,\delta _{n,n'}\,\delta _{k,k'}}$

${\displaystyle V(\rho ,\varphi ,z)=\sum _{n,k}A_{n,k}V_{n,k}}$ ，k取分立值
${\displaystyle V(\rho ,\varphi ,z)=\sum _{n}\int dk\,A_{n}(k)V_{n,k}}$ ，k取連續值