泽尔尼克多项式

泽尔尼克多项式是一个以1953年获诺贝尔物理学奖荷兰物理学家弗里茨·泽尔尼克命名的正交多项式，分为奇、偶两类 奇多项式： Z n m ( ρ , φ ) = R n m ( ρ ) cos ⁡ ( m φ ) {\displaystyle Z_{n}^{m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\cos(m\,\varphi )\!} 偶多项式 Z n − m ( ρ , φ ) = R n m ( ρ ) sin ⁡ ( m φ ) , {\displaystyle Z_{n}^{-m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\sin(m\,\varphi ),\!} 其中 n ≥ m {\displaystyle n\geq m} 为非负整数， ϕ {\displaystyle \phi } 为方位角 0 ≤ ρ ≤ 1 {\displaystyle 0\leq \rho \leq 1} 为径向距离 如果 n-m为偶数则 R n m ( ρ ) = ∑ k = 0 n − m 2 ( − 1 ) k ( n − k ) ! k ! ( n + m 2 − k ) ! ( n − m 2 − k ) ! ρ n − 2 k {\displaystyle R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}{\frac {(-1)^{k}\,(n-k)!}{k!\left({\tfrac {n+m}{2}}-k\right)!\left({\tfrac {n-m}{2}}-k\right)!}}\;\rho ^{n-2\,k}} 如果n-m为奇数，则 R n m ( ρ ) = 0 {\displaystyle R_{n}^{m}(\rho )=0} 泽尔尼克多项式也可以表示为超几何函数 R n m ( ρ ) = ( n n + m 2 ) ρ n   2 F 1 ( − n + m 2 , − n − m 2 ; − n ; ρ − 2 ) = ( − 1 ) n + m 2 ( n + m 2 n − m 2 ) ρ m   2 F 1 ( 1 + n , 1 − n − m 2 ; 1 + n + m 2 ; ρ 2 ) {\displaystyle {\begin{aligned}R_{n}^{m}(\rho )&={\binom {n}{\tfrac {n+m}{2}}}\rho ^{n}\ {}_{2}F_{1}\left(-{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};-n;\rho ^{-2}\right)\\&=(-1)^{\tfrac {n+m}{2}}{\binom {\tfrac {n+m}{2}}{\tfrac {n-m}{2}}}\rho ^{m}\ {}_{2}F_{1}\left(1+n,1-{\tfrac {n-m}{2}};1+{\tfrac {n+m}{2}};\rho ^{2}\right)\end{aligned}}} Noll 用一个J数字表示 [n,m]:如下表 由于 I j = ∫ 0 2 π ∫ 0 1 Z j 2 ρ d ρ d θ = k j ∗ π . {\displaystyle I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =k_{j}*\pi .} 其中 k j {\displaystyle k_{j}} 因j而异， k 1 = 1 {\displaystyle k_{1}=1} k 2 = 1 4 {\displaystyle k_{2}={\frac {1}{4}}} k 3 = 1 4 {\displaystyle k_{3}={\frac {1}{4}}} k 4 = 1 3 {\displaystyle k_{4}={\frac {1}{3}}} k 5 = 1 6 {\displaystyle k_{5}={\frac {1}{6}}} 必须先归一化 令 Z j = Z j / ( k j ) {\displaystyle Z_{j}=Z_{j}/{\sqrt {(}}k_{j})} 使得 I j = ∫ 0 2 π ∫ 0 1 Z j 2 ρ d ρ d θ = π . {\displaystyle I_{j}=\int _{0}^{2\pi }\int _{0}^{1}Z_{j}^{2}\,\rho \,d\rho \,d\theta =\pi .} 归一化泽尔尼克多项式以Noll序列排列如下： 径向正交性 ∫ 0 1 ρ 2 n + 2 R n m ( ρ ) 2 n ′ + 2 R n ′ m ( ρ ) d ρ = δ n , n ′ . {\displaystyle \int _{0}^{1}\rho {\sqrt {2n+2}}R_{n}^{m}(\rho )\,{\sqrt {2n'+2}}R_{n'}^{m}(\rho )\,d\rho =\delta _{n,n'}.} 角度正交性 ∫ 0 2 π cos ⁡ ( m φ ) cos ⁡ ( m ′ φ ) d φ = ϵ m π δ | m | , | m ′ | , {\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\cos(m'\varphi )\,d\varphi =\epsilon _{m}\pi \delta _{|m|,|m'|},} ∫ 0 2 π sin ⁡ ( m φ ) sin ⁡ ( m ′ φ ) d φ = ( − 1 ) m + m ′ π δ | m | , | m ′ | ; m ≠ 0 , {\displaystyle \int _{0}^{2\pi }\sin(m\varphi )\sin(m'\varphi )\,d\varphi =(-1)^{m+m'}\pi \delta _{|m|,|m'|};\quad m\neq 0,} ∫ 0 2 π cos ⁡ ( m φ ) sin ⁡ ( m ′ φ ) d φ = 0 , {\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,} 其中 ϵ m {\displaystyle \epsilon _{m}} 称为Neumann因子，其数值为 2 如果满足 m = 0 {\displaystyle m=0} ，数值为 1，如果 m ≠ 0 {\displaystyle m\neq 0} . 径向与角度正交性 ∫ Z n m ( ρ , φ ) Z n ′ m ′ ( ρ , φ ) d 2 r = ϵ m π 2 n + 2 δ n , n ′ δ m , m ′ , {\displaystyle \int Z_{n}^{m}(\rho ,\varphi )Z_{n'}^{m'}(\rho ,\varphi )\,d^{2}r={\frac {\epsilon _{m}\pi }{2n+2}}\delta _{n,n'}\delta _{m,m'},} 其中 d 2 r = ρ d ρ d φ {\displaystyle d^{2}r=\rho \,d\rho \,d\varphi } 为 雅可比矩阵 n − m {\displaystyle n-m} 与 n ′ − m ′ {\displaystyle n'-m'} 都是偶数. 埃里克·韦斯坦因. 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