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Recurrence formulae for Rn,m and Sn,m

The associated Legendre function Pnm has the following recurrence formulae:
  
$\displaystyle (n-m+1)P_{n+1}^m(\cos \theta) - (2n+1) \cos \theta P_n^m(\cos \theta)
+(n+m)P_{n-1}^m(\cos \theta) = 0,$     (C.1)
$\displaystyle \cos\theta P_n^m(\cos \theta) - P_{n+1}^m(\cos \theta)
+(n+m)\sin\theta P_{n,m-1}(\cos \theta) = 0.$     (C.2)

Multiplying (C.1) by $r^{n+1}e^{im\phi}$ one obtains (n+m+1)(n+1-m)Rn+1,m - (2n+1) x3 Rn,m + r2 Rn-1,m=0.

Multiplying (C.2) by $r^{n+1}e^{im\phi}$, setting m=n+1 and using (A.2) one obtains

\begin{eqnarray*}R_{n+1,n+1} = \frac{x_1 + i x_2}{2(n+1)}R_{n,n}.
\end{eqnarray*}


In the same manner, one has r2 Sn+1,m-(2n+1) x3 Sn,m + (n+m)(n-m)Sn-1,m=0 .


\begin{eqnarray*}S_{n+1,n+1} = \frac{(2n+1)(x_1 + i x_2)}{r^2}S_{n,n}.
\end{eqnarray*}


Using above formulae one can compute Rn,m and Sn,m as follows:


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