next up previous contents
Next: Relations between solid harmonics Rn,m Sn,m Up: Applications of Fast Multipole Previous: Final remarks

   
Series expansion of $\frac{1}{\vert x -y\vert}$

Now we take two points $\mbox{\boldmath$\space y $ }$ and $\mbox{\boldmath$\space x $ }$ whose spherical coordinates are $(\rho,\alpha,\beta)$ and $(r,\phi,\theta)$. The inverse of the distance between $\mbox{\boldmath$\space y $ }$ and $\mbox{\boldmath$\space x $ }$ can be expanded into the following series[61]
 
$\displaystyle \frac{1}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=0}^{n}\varepsilon_m
\frac{(n-m)!}{(n+m)!} P_n^m(\cos\alpha) P_n^m(\cos\theta)
\cos m(\beta-\phi) \frac{\rho^n}{r^{n+1}},$ (A.1)

where $\varepsilon_m$ is the Neumann factor; $\varepsilon_0 = 1,
\varepsilon_i=2(i=1,2,3,\ldots)$ and Pnm(x) is the associated Legendre function. In this thesis the associated Legendre function is defined by

\begin{eqnarray*}P_n^m(x) = (1-x^2)^{m/2}\frac{d^m}{dx^m} P_n(x)\quad (m \ge 0),
\end{eqnarray*}


where Pn(x) is the Legendre function and the associated Legendre function has the following properties:
   
$\displaystyle P_n^{m}(\cos\theta)$ = $\displaystyle 0,\quad (n < \vert m\vert)$ (A.2)
$\displaystyle P_n^{-m}(\cos\theta)$ = $\displaystyle (-1)^m\frac{(n-m)!}{(n+m)!}P_n^m(\cos\theta)$ (A.3)
$\displaystyle P_n^m(-\cos\theta)$ = $\displaystyle (-1)^{n+m}P_n^m(\cos\theta)$ (A.4)

The right-hand side in (A.1) can be rewritten as

\begin{eqnarray*}\frac{1}{\vert\mbox{\boldmath$ x $ }- \mbox{\boldmath$ y $ }\ve...
...$ y $ }})S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})\qquad
\end{eqnarray*}


where we have used (A.3) and functions $R_{n,m}(\overrightarrow{O\mbox{\boldmath$\space y $ }})$ and $S_{n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are defined by

\begin{eqnarray*}R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})&=&\frac{1}{(n...
...th$ x $ }})&=&(n-m)!P_n^m(\cos\theta)e^{im\phi}\frac{1}{r^{n+1}}
\end{eqnarray*}


The solid harmonics Rn,m and Sn,m have the following properties.

 
$\displaystyle R_{n,-m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{1}{(n-m)!}P_n^{-m}(\cos\theta)e^{-im\phi}r^n
=\frac{1}{(n-m)!}(-1)^m\frac{(n-m)!}{(n+m)!}
P_n^m(\cos\theta)e^{-im\phi}r^n$  
  = $\displaystyle (-1)^m\frac{1}{(n+m)!}P_n^m(\cos\theta)e^{-im\phi}r^n
=(-1)^m\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ (A.5)


 
$\displaystyle S_{n,-m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{(n+m)!}{r^{n+1}}P_n^{-m}(\cos\theta)e^{-im\phi}
=\frac{(n+m)!}{r^{n+1}}(-1)^m\frac{(n-m)!}{(n+m)!}
P_n^m(\cos\theta)e^{-im\phi}$  
  = $\displaystyle (-1)^m\frac{(n-m)!}{r^{n+1}}P_n^m(\cos\theta)e^{-im\phi}
=(-1)^m\overline{S_{n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ (A.6)

where we have used (A.3), and
  
$\displaystyle R_{n,m}(\overrightarrow{\mbox{\boldmath$ x $ }O})$ = $\displaystyle \frac{1}{(n+m)!}P_n^m(\cos(\theta+\pi))e^{im(\phi+\pi)}r^n
=\frac{1}{(n+m)!}P_n^m(-\cos\theta)(-1)^m e^{im\phi}r^n$  
  = $\displaystyle \frac{1}{(n+m)!}(-1)^{n+m}P_n^m(\cos\theta)(-1)^m e^{im\phi}r^n
=(-1)^n R_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ (A.7)
$\displaystyle S_{n,m}(\overrightarrow{\mbox{\boldmath$ x $ }O})$ = $\displaystyle \frac{(n-m)!}{r^{n+1}}P_n^m(\cos(\theta+\pi))
e^{im(\phi+\pi)} =\frac{(n-m)!}{r^{n+1}}P_n^m(-\cos\theta)
(-1)^m e^{im\phi}$  
  = $\displaystyle \frac{(n-m)!}{r^{n+1}}(-1)^{n+m}P_n^m(\cos\theta)(-1)^m e^{im\phi}
=(-1)^n S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ (A.8)

where we have used (A.4).


next up previous contents
Next: Relations between solid harmonics Rn,m Sn,m Up: Applications of Fast Multipole Previous: Final remarks
Ken-ichi Yoshida
2001-07-28