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Derivatives of Rn,m and Sn,m

We introduce a vector $\overrightarrow{O\mbox{\boldmath$\space x $ }}$ which has components (x1,x2,x3) in the Cartesian coordinates and $(r,\phi,\theta)$ in the polar coordinates. Relations between (x1,x2,x3) and $(r,\phi,\theta)$ are expressed as

\begin{eqnarray*}x_1=r \cos\phi \sin\theta, \quad x_2=r \sin\phi \sin\theta, \quad x_3= r \cos\theta,
\end{eqnarray*}



\begin{eqnarray*}r=\sqrt{x_1^2+x_2^2+x_3^2},\quad \phi=\tan^{-1}\frac{x_2}{x_1},\quad \theta=\tan^{-1}\frac{\sqrt{x_1^2+x_2^2}}{x_3}.
\end{eqnarray*}


According to the chain rule, we have the following relations:

\begin{eqnarray*}\left(\frac{\partial}{\partial x_1} + i \frac{\partial}{\partia...
...l \theta}{\partial x_2} \right)\frac{\partial}{\partial \theta},
\end{eqnarray*}


the coefficients of which are given as

\begin{eqnarray*}\left(\frac{\partial r}{\partial x_1}
+ i \frac{\partial r}{\p...
...l \theta}{\partial x_2} \right) = \frac{\cos\theta}{r}e^{i\phi}.
\end{eqnarray*}


Now we compute $\left(\frac{\partial}{\partial x_1} + i
\frac{\partial}{\partial x_2}\right)R_{n,m}$ for example.
 
    $\displaystyle (n+m)!\left(\frac{\partial}{\partial x_1} + i \frac{\partial}{\partial x_2}\right)R_{n,m}$  
    $\displaystyle = \left( \left(\frac{\partial r}{\partial x_1}
+ i \frac{\partial...
...ight)\frac{\partial}{\partial \theta} \right) (r^n P_n^m(\cos\theta)e^{im\phi})$  
    $\displaystyle = \sin\theta n r^{n-1}P_n^m(\cos\theta) e^{i(m+1)\phi}
- \frac{1}...
...\theta r^{n-1} \frac{\partial P_n^m(\cos\theta)}{\partial \theta}e^{i(m+1)\phi}$  
    $\displaystyle = - r^{n-1} P_{n-1}^{m+1} e^{i(m+1)\phi}
\Rightarrow \left(\frac{...
...\partial x_1} + i \frac{\partial}{\partial x_2}\right) R_{n,m} = - R_{n-1,m+1},$ (D.1)

where we have used standard formulae given by

\begin{eqnarray*}\frac{\partial P_n^m(\cos\theta)}{\partial \theta}=-P_n^{m+1}(\...
...ta)-\cos\theta P_n^{m+1}(\cos\theta)=-P_{n-1}^{m+1}(\cos\theta).
\end{eqnarray*}


In the same way we obtain formulae given by
  
$\displaystyle \left(\frac{\partial}{\partial x_1} - i \frac{\partial}{\partial x_2}\right)
R_{n,m}$ = Rn-1,m-1, (D.2)
$\displaystyle \frac{\partial}{\partial x_3} R_{n,m}$ = Rn-1,m. (D.3)

We can rewrite (D.1), (D.2), (D.3) as
$\displaystyle \frac{\partial}{\partial x_1}R_{n,m}$ = $\displaystyle \frac{1}{2}(R_{n-1,m-1}-R_{n-1,m+1}) ,$ (D.4)
$\displaystyle \frac{\partial}{\partial x_2}R_{n,m}$ = $\displaystyle \frac{i}{2}(R_{n-1,m-1}+R_{n-1,m+1}) ,$ (D.5)
$\displaystyle \frac{\partial}{\partial x_3} R_{n,m}$ = Rn-1,m. (D.6)

Derivatives of Sn,m can be obtained in the similar manner as follows:
   
$\displaystyle \left(\frac{\partial}{\partial x_1} + i \frac{\partial}{\partial x_2}\right) S_{n,m}$ = - Sn+1,m+1, (D.7)
$\displaystyle \left(\frac{\partial}{\partial x_1} - i \frac{\partial}{\partial x_2}\right) S_{n,m}$ = Sn+1,m-1, (D.8)
$\displaystyle \frac{\partial}{\partial x_1}S_{n,m}$ = $\displaystyle \frac{1}{2}(S_{n+1,m-1}-S_{n+1,m+1}) ,$ (D.9)
$\displaystyle \frac{\partial}{\partial x_2}S_{n,m}$ = $\displaystyle \frac{i}{2}(S_{n+1,m+1}+S_{n+1,m-1}) ,$ (D.10)
$\displaystyle \frac{\partial}{\partial x_3}S_{n,m}$ = - Sn+1,m. (D.11)


next up previous contents
Next: M2M, M2L, L2L in Up: Applications of Fast Multipole Previous: Recurrence formulae for Rn,m Sn,m
Ken-ichi Yoshida
2001-07-28